Output feedback control of radially-dependent reaction-diffusion PDEs on balls of arbitrary dimensions

IFAC PapersOnLine 53-2 (2020) 7635–7640 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2020 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2020.12.1364 10.1016/j.ifacol.2020.12.1364 2405-8963 Copyright © 2020 The Authors. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0 ) Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and jie[email protected]du.cn). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]du). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). Output Feedback Control of Radially-Dependent Reaction-Diffusion PDEs on Balls of Arbitrary Dimensions Rafael Vazquez ∗Jing Zhang ∗∗ Miroslav Krstic ∗∗∗ Jie Qi ∗∗ ∗Dept. of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimiento s.n., 41092 Sevilla, Spain (e-mail: [email protected]). ∗∗ College of Information Science and Technology, Donghua University, Shanghai 201620 China (e-mails: [email protected]om and [email protected]). ∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA (e-mail: [email protected]). Abstract: Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, wellposedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems. 1. INTRODUCTION In this paper we introduce an explicit boundary outputfeedback control law to stabilize an unstable linear radially-dependent reaction-diffusion equation on an n-ball (which in 2-D is a disk and in 3-D a sphere). This paper extends the spherical harmonics [4] approach of [20], which assumed constant coefficients, using some of the ideas of [24]; for the sake of brevity we will mainly show the modifications required with respect to [20], skipping the details when they are identical. For a finite number of harmonics, we design boundary feedback laws and output injection gains using the backstepping method [7] (with kernels computed using a power series approach) which allows us to obtain exponential stability of the origin in the L2norm. Higher harmonics will be naturally open-loop stable. The required conditions for the radially-varying coefficients are found in the analysis of the numerical method and are non-obvious (evenness of the reaction coefficient). The idea of using a power series to compute backstepping This work was supported by the National Natural Science Foundation of China (61773112), the Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D-2019089) and the scholarship from China Scholarship Council (CSC201806630010). R. Vazquez acknowledges financial support of the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-100680-B-C21 kernels was first seen in [2] (without much analysis of the method itself, but rather numerically optimizing the approximation) and later in [6], where piecewise-smooth kernels require the use of several series. Here, we state that the method provides a unique solution leaving the question of convergence of the series (i.e., analyticity of the kernel) for future work; the proof of the result is skipped due to page limitation and will be given in an upcoming publication. Some partial results towards the solution of this problem were obtained in [22] and [21] for the disk and sphere, respectively; however they required simmetry conditions. Older results in this spirit were obtained in [18] and [12]. Previous results and applications in multi-dimensional domains include multi-agent deployment in 3-D space [13] (by combining the ideas of [20] and [10]), convection problem on annular domains [19], PDEs with boundary conditions governed by lower-dimensional PDEs [14, 24], multi-dimensional cuboid domains [11]. The backstepping method has proved itself to be an ubiquitous method for PDE control, with many other applications including, among others, flow control [16, 23], nonlinear PDEs [17], hyperbolic 1-D systems [9, 3, 1], or delays [8]. Nevertheless, other design methods are also applicable to the geometry considered in this paper (see for instance [15] or [5]). 7636 Rafael Vazquez et al. / IFAC PapersOnLine 53-2 (2020) 7635–7640 The structure of the paper is as follows. In Section 2 we introduce the problem. In Section 3 we state our stability result. We state the well-posedness of the kernels in Section 4, but the proof is skipped due to page limitation. We briefly talk about the observer in Section 5, but skip most details based on its duality with respect to the controller. Next we give some simulation results in Section 6. We finally conclude the paper with some remarks in Section 7. 2. N-D REACTION-DIFFUSION SYSTEM ON AN N-BALL Following [20], a varying coefficient reaction-diffusion system in an n-dimensional ball of radius Rcan be written in n-dimensional spherical coordinates, also known as ultraspherical coordinates (see [4], p. 93), which consist of one radial coordinate and n−1angular coordinates. Then, using a (complex-valued) Fourier-Laplace series of Spherical Harmonics 1to handle the angular dependencies, one reaches the following independent complex-valued 1-D reaction-diffusion equation for each harmonic: ∂tum l= rn−1∂rrn−1∂rum l−l(l+n−2)  r2um l+λ(r)um l, (1) evolving in r∈[0,R],t>0, with boundary conditions um l(t, R)=Um l(t),(2) In these equations, we have considered Dirichlet boundary conditions. The measurement would be the flux at the boundary, namely ∂rum l(t, R). In the above equations, the integers mand lstand for the order and degree of the harmonic, respectively. Note that the higher the degree (corresponding to high frequencies), the more “naturally” stable (1)–(2) is, as seen next. Define the L2norm f(r)2 L2= R  0 |f(r)|2rn−1dr. (3) and the associated L2space as usual, where |f|2=ff∗, being f∗the complex conjugate of f. Lemma 1. Given λ(r)and R, there exists L∈Nsuch that, for all l>L, the equilibrium um l≡0of system (1)-(2) is open loop exponentially stable, namely, for Um l=0in (1) there exists a positive constant D1, such that for all t um l(t, ·)2 L2≤e−D1tum l(0,·)2 L2.(4) D1is independent of l, and only depends on n,λ(r),, and R, and can be chosen as large as desired just by increasing the values of L. The proof is skipped as it mimics [24] just by using the L2 norm as a Lyapunov function and Poincare’s inequality. Thus one only needs to stabilize the unstable mode with |l|<L. Since the different modes are not coupled, it allows us to stabilize them separately and re-assembling them. Moreover since only a finite number of harmonics is stabilized, there is no need to worry about the convergence of the control law as in [20], with its Spherical Harmonics series being just a finite sum. 1Spherical harmonics were introduced by Laplace to solve the homonymous equation and have been widely used since, particularly in geodesics, electromagnetism and computer graphics. A very complete treatment on the subject can be found in [4]. Our objective can now be stated as follows. Considering only the unstable modes, design an output-feedback control law for Um lusing, for each mode, only the measurement of ∂rum l(t, R). Our design procedure is established in the next section along with our main stability result. 3. STABILITY OF CONTROLLED HARMONICS Next, for the unstable modes we design the outputfeedback law. The observer and controller are designed separately using the backstepping method, by following [20]; in this reference it is shown that both the feedback and the output injection gains can be found by solving a certain kernel PDE equation, which is essentially the same for both the controller and the observer. Thus, for the sake of brevity and to avoid repetitive material, we only show how to obtain the (full-state) control law, giving the basic observer design and some additional remarks later in Section 5. 3.1 Design of a full-state feedback control law for unstable modes Based on the backstepping method [7], our idea is utilizing a invertible Volterra integral transformation wm l(t, r)=um l(t, r)− r  0 Kn lm(r, ρ)um l(t, ρ)dρ, (5) where the kernel Kn lm(r, ρ)is to be determined, which defined on the domain Tk={(r, ρ)∈R2;0 ≤ρ≤ r≤R}to convert the unstable system (1)-(2) into an exponentially target system: ∂twm l=∂r(rn−1∂rwm l) rn−1−l(l+n−2)wm l r2−cwm l,(6) wm l(t, R)=0,(7) where the constant c>0is an adjustable convergence rate. From (5) and (7), let r=R, we obtain the boundary control as the following full-state law Um l(t)= R  0 Kn lm(R, ρ)um l(t, ρ)dρ. (8) Following closely the steps of [20] to find conditions for the kernels, and defining Kn lm(r, ρ)=Gn lm(r, ρ)ρρ rl+n−2, we finally reach a PDE that the G-kernels need to verify: λ(ρ)+c Gn lm =∂rrGn lm + (3 −n−2l)∂rGn lm r −∂ρρGn lm + (1 −n−2l)∂ρGn lm ρ,(9) with only one boundary condition: Gn lm(r, r)=−r 0(λ(σ)+c)dσ 2r .(10) We assume as usual that these kernel equations are wellposed and the resulting kernel is bounded in T; this will be analyzed later in Section 4, providing a numerical method for its computation. 3.2 Closed-loop stability analysis of unstable modes To obtain the stability result of closed-loop system, we need three elements. We begin by stating the stability Rafael Vazquez et al. / IFAC PapersOnLine 53-2 (2020) 7635–7640 7637 The structure of the paper is as follows. In Section 2 we introduce the problem. In Section 3 we state our stability result. We state the well-posedness of the kernels in Section 4, but the proof is skipped due to page limitation. We briefly talk about the observer in Section 5, but skip most details based on its duality with respect to the controller. Next we give some simulation results in Section 6. We finally conclude the paper with some remarks in Section 7. 2. N-D REACTION-DIFFUSION SYSTEM ON AN N-BALL Following [20], a varying coefficient reaction-diffusion system in an n-dimensional ball of radius Rcan be written in n-dimensional spherical coordinates, also known as ultraspherical coordinates (see [4], p. 93), which consist of one radial coordinate and n−1angular coordinates. Then, using a (complex-valued) Fourier-Laplace series of Spherical Harmonics 1to handle the angular dependencies, one reaches the following independent complex-valued 1-D reaction-diffusion equation for each harmonic: ∂tum l= rn−1∂rrn−1∂rum l−l(l+n−2)  r2um l+λ(r)um l, (1) evolving in r∈[0,R],t>0, with boundary conditions um l(t, R)=Um l(t),(2) In these equations, we have considered Dirichlet boundary conditions. The measurement would be the flux at the boundary, namely ∂rum l(t, R). In the above equations, the integers mand lstand for the order and degree of the harmonic, respectively. Note that the higher the degree (corresponding to high frequencies), the more “naturally” stable (1)–(2) is, as seen next. Define the L2norm f(r)2 L2= R  0 |f(r)|2rn−1dr. (3) and the associated L2space as usual, where |f|2=ff∗, being f∗the complex conjugate of f. Lemma 1. Given λ(r)and R, there exists L∈Nsuch that, for all l>L, the equilibrium um l≡0of system (1)-(2) is open loop exponentially stable, namely, for Um l=0in (1) there exists a positive constant D1, such that for all t um l(t, ·)2 L2≤e−D1tum l(0,·)2 L2.(4) D1is independent of l, and only depends on n,λ(r),, and R, and can be chosen as large as desired just by increasing the values of L. The proof is skipped as it mimics [24] just by using the L2 norm as a Lyapunov function and Poincare’s inequality. Thus one only needs to stabilize the unstable mode with |l|<L. Since the different modes are not coupled, it allows us to stabilize them separately and re-assembling them. Moreover since only a finite number of harmonics is stabilized, there is no need to worry about the convergence of the control law as in [20], with its Spherical Harmonics series being just a finite sum. 1Spherical harmonics were introduced by Laplace to solve the homonymous equation and have been widely used since, particularly in geodesics, electromagnetism and computer graphics. A very complete treatment on the subject can be found in [4]. Our objective can now be stated as follows. Considering only the unstable modes, design an output-feedback control law for Um lusing, for each mode, only the measurement of ∂rum l(t, R). Our design procedure is established in the next section along with our main stability result. 3. STABILITY OF CONTROLLED HARMONICS Next, for the unstable modes we design the outputfeedback law. The observer and controller are designed separately using the backstepping method, by following [20]; in this reference it is shown that both the feedback and the output injection gains can be found by solving a certain kernel PDE equation, which is essentially the same for both the controller and the observer. Thus, for the sake of brevity and to avoid repetitive material, we only show how to obtain the (full-state) control law, giving the basic observer design and some additional remarks later in Section 5. 3.1 Design of a full-state feedback control law for unstable modes Based on the backstepping method [7], our idea is utilizing a invertible Volterra integral transformation wm l(t, r)=um l(t, r)− r  0 Kn lm(r, ρ)um l(t, ρ)dρ, (5) where the kernel Kn lm(r, ρ)is to be determined, which defined on the domain Tk={(r, ρ)∈R2;0 ≤ρ≤ r≤R}to convert the unstable system (1)-(2) into an exponentially target system: ∂twm l=∂r(rn−1∂rwm l) rn−1−l(l+n−2)wm l r2−cwm l,(6) wm l(t, R)=0,(7) where the constant c>0is an adjustable convergence rate. From (5) and (7), let r=R, we obtain the boundary control as the following full-state law Um l(t)= R  0 Kn lm(R, ρ)um l(t, ρ)dρ. (8) Following closely the steps of [20] to find conditions for the kernels, and defining Kn lm(r, ρ)=Gn lm(r, ρ)ρρ rl+n−2, we finally reach a PDE that the G-kernels need to verify: λ(ρ)+c Gn lm =∂rrGn lm + (3 −n−2l)∂rGn lm r −∂ρρGn lm + (1 −n−2l)∂ρGn lm ρ,(9) with only one boundary condition: Gn lm(r, r)=−r 0(λ(σ)+c)dσ 2r .(10) We assume as usual that these kernel equations are wellposed and the resulting kernel is bounded in T; this will be analyzed later in Section 4, providing a numerical method for its computation. 3.2 Closed-loop stability analysis of unstable modes To obtain the stability result of closed-loop system, we need three elements. We begin by stating the stability result for the target system. We follow by obtaining the existence of an inverse transformation that allows us to recover our original variable from the transformed variable. Then we relate the L2norm with spherical harmonics. With these elements, we construct the proof of stability mapping the result for the target system to the original system. This is done by showing that the transformation is an invertible map from L2into L2. We first discuss the stability of the target system, having the following Lemma: Lemma 2. For all l∈N, and for c≥0, the equilibrium wm l≡0of system (6)–(7) is exponentially stable, i.e., there exists a positive constant D2such that for all t, wm l(t, ·)2 L2≤e−D2twm l(0,·)2 L2,(11) where the constant D2is independent of n,lor m, and only depends on c,, and R; it can be chosen as large as desired just by increasing the value of c. Proof. Consider the Lyapunov function: V2(t)=1 2wm l(t, ·)2 L2,(12) then, taking its time derivative, we obtain ˙ V2= R  0 ¯wm l∂twm l+wm l∂t¯wm l 2rn−1dr ≤− 4R2+cwm l2 L2(13) choosing c=D2 2− 4R2(14) we then obtain, independent of the value of n, ˙ V2≤−D2V2,(15) thus proving the result. Lemma 3. For |l|≤ L, let cbe chosen as in Lemma 2, and assume that the kernel Kn lm(r, ρ)is bounded and integrable. The system (1) with boundary control (8) is closed-loop exponentially stable, namely there exists positive constants Cand D2such that um l(t, ·)2 L2≤Ce−D2tum l(0,·)2 L2,(16) Cand D2are independent of mor l, and only depend on n,L,λ(r),and R. Proof. The proof consists of two parts, one is existence of an inverse transformation, and then showing the equivalence of norms of the variables un lm and wn lm; the result then follows from the stability of the target system. As shown in [20], when Kn(r, ρ)is bounded and integrable, the map (5) is reversible and its inverse transformation is um l(t, r)=wm l(t, r)+ r  0 Ln lm(r, ρ)wm l(t, ρ)dρ, (17) which is also bounded and integrable. Call now ¯ Kand ¯ L the maximum of the bounds of the function ˇ Kn lm and ˇ Ln lm for a given nand all |l|≤Lin their respective domains. It is easy to get wm l(t, ·)2 L2≤M1um l(t, ·)2 L2,(18) um l(t, ·)2 L2≤M2wm l(t, ·)2 L2.(19) where M1= 2+R4¯ K/(2n)and M2= 2+R4¯ L/(2n). Combining then Lemma 2 with the norm equivalence between um land wm lsystem stated as in (18) and (19), it is easy to obtain um l(t, ·)2 L2≤M2wm l(t, ·)2 L2 ≤M2e−D2twm l(0,·)2 L2≤M1M2e−D2tum l(0,·)2 L2.(20) Let C=M1M2, the result then follows.  Note that combining Lemmas 1 and 3 and taking D= min{D1,D 1}, we get the following stability result for all spherical harmonics and thus the full physical system. Theorem 4. The equilibrium um l≡0of system (1)-(2) under control law (8) is closed-loop exponentially stable, namely, there exists a positive constant D, such that for all t um l(t, ·)2 L2≤Ce−Dtum l(0,·)2 L2.(21) where Dcan be chosen as large as desired just by increasing the value of Land cin the control design process. 4. WELL-POSEDNESS OF THE KERNEL EQUATIONS Next, we state a kernell well-posedness result, which was in part implicitly assumed in Theorem 4, also giving the requirements for λ(r). This result also provides a numerical method to compute the kernels, which is an alternative to successive approximations which do not work so well in this case (due to the singularities present at the origin; see for instance [22] to see the resulting singular integral equation that needs to be solved). Theorem 5. Under the assumption that λ(r)is an even analytic function in [0,1], then for a given n>1and all values of l∈N, there is a unique power series solution Gn lm(r, ρ)for (9)–(10), even in its two variables in the domain T. In addition, if λ(r)is analytic but not even, then there is no power series solution to (9)–(10) for all l∈N. Remark 1. The requirement of evenness for λ(r)might seem strange at first sight. However, note that r= x2 1+x2 2+...+x2 n, therefore in physical space λ(r)will be non-smooth unless it is even. Thus, while the kernels might exist for non-even λ(r), we cannot expect them to be smooth, which might be indeed problematic for higherorder harmonics; not so much for lower-order, as shown in [22], which only considers the 0-th order harmonic and consequently only requires boundedness of λ(r). The proof of Theorem 5 cannot be given due to page limitation and will appear in a subsequent journal publication. It is based on expressing a solution to (9)–(10) in the form: Gn lm(r, ρ)= ∞  i=0   i  j=0 Cijrjρi−j ,(22) Note that the series in (22) has been written in a way that collects together (in the parenthesis) all the polynomial terms with the same degree. 5. OBSERVER DESIGN This section designs an observer for (1)-(2) from the measured output ∂rum l(t, R)as follows: 7638 Rafael Vazquez et al. / IFAC PapersOnLine 53-2 (2020) 7635–7640 ∂tˆum l=∂rrn−1∂rˆum l rn−1−l(l+n−2)  r2ˆum l+λ(r)ˆum l +pn lm(r)(∂rum l(t, R)−∂rˆum l(t, R)),(23) with boundary condition ˆ Um l(t)=Um l(t).(24) We need to design the output injection gain pn lm(r). Closely following [20], define the observer error as ˜u=u−ˆu. The observer error dynamics are given by ∂˜ulmt ∂t = rn−1∂rrn−1∂r˜ulm−l(l+n−2)  r2˜ulm +λ(r)˜ulm −pn lm(r)∂r˜ulm(t, R),(25) with boundary conditions ˜ulm(t, R)=0.(26) Next we use the backstepping method to find a value of pn lm(r)that guarantees convergence of ˜uto zero. This ensures that the observer estimates tend to the true state values. Our approach to design p(r)is to seek a mapping that transforms (25) into the following target system ∂˜wlmt ∂t = rn−1∂rrn−1∂r˜wlm −c˜wlm −l(l+n−2)  r2˜wlm,(27) with boundary conditions ˜wlm(t, R)=0.(28) The transformation is defined as follows: ˜ulm(t, r)= ˜wlm(t, r)−R r Pn lm(r, ρ)˜wlm(t, ρ)dρ, (29) and then pn lm(r)will be found from transformation kernel as an additional condition. From [20], one obtains the following PDE that the kernel must verify: 1 rn−1∂rrn−1∂rPn lm−∂ρρn−1∂ρPn lm ρn−1 −l(l+n−2) 1 r2−1 ρ2Pn lm =−λ(r) Pn lm (30) In addition we find a value for the output injection gain kernel pn lm(r)=Pn lm(r, R)(31) Also, the following boundary condition has to be verified 0=λ(r)+(∂rPn lm(r, ρ)) ρ=r + rn−1 d dr rn−1Pn lm(r, r) +∂ρPn lm(r, ρ) ρn−1ρ=r rn−1,(32) which can be written as 0=λ(r)+∂rPn lm(r, r)+d dr (Pn lm(r, r))+(n−1)Pn lm(r, r) r +∂ρPn lm(r, r)−(n−1)P n lm(r, r) r.(33) Following [20], and after some computations, we reach boundary conditions for the kernel equations as follows Pn lm(0,ρ)=0,∀l=0 (34) Pn lmr(0,ρ)=0,∀l=1 (35) Pn lm(r, r)=−r 0λ(σ)dσ 2.(36) It turns out the observer kernel equation can be transformed into the control kernel equation, therefore obtaining a similar explicit result. For this, define ˇ Pn lm(r, ρ)=ρn−1 rn−1Pn lm(ρ, r),(37) and it can be verified that the equation now verified by ˇ Pn lm(r, ρ)is exactly the equation verified by Kn lm(r, ρ). Thus ˇ Pn lm(r, ρ)=Kn lm(r, ρ)and we can apply our previous result of Section 4. The observer error ˜uhas the same stability properties derived in Section 3 for the closed-loop system under the full-state control. As in the controller case, only a limited number of modes need to be estimated; namely, those that are not naturally stable by virtue of Lemma 2, this being the main difference with the result given in [20]. Finally, the controller-observer augmented system can be proved closed-loop stable as in [20], using the separation principle given the linearity of the system, with desired convergence rate, and without much modification; we skip the details for lack of space, which requires going up to H1 stability, as in [20]. 6. SIMULATION STUDY In this section, the simulation experiment on threedimensional ball (n=3) is taken as an example to illustrate the effectiveness of proposed control. The system with the output-feedback control law is simulated over 0≤t≤2s with the following parameters: =1,λ(r) = 10r4+50r2+ 50,c=3. We consider that the system is initially at the random quantity, u0∈[0,10], and the observer’s initial condition is set as actual state plus an error of normal distribution with zero mean and σ2=0.5. Fig. 1 shows the plots of the polynomial approximation of kernels K3 lm, which is obtained by using (22) up to a cutoff at the p-th powers. The value of Kdoes not depend on mso we omit that sub-index. The value of pis chosen as p= 15. Applying Lemma 1, one can obtain Lto be 11; however, here to save space, we only show the first six approximate numerical solutions of control gains. As shown in Fig. 1, we find that the Klbecomes increasingly smaller when lincreases. In order to avoid a dramatic increase in the complexity of simulation caused by the high dimension, in our simulations we employ a method also based on spherical harmonics expansions which greatly reduces the error. Thus, we only calculate the harmonics ulm which only need discretization in the radial direction, and then sum a finite number Sof harmonics to recover u. When S>0 is a large enough integer, the error caused by the use of a finite number of harmonics is much smaller than the angular discretization error. Thus, the simulation is carried out using the formula u(t, r, θ1,θ 2)= l=S  l=0 m=l  m=−l um l(t, r)Y3 lm(θ1,θ 2)(38) where the spherical harmonics are defined as Rafael Vazquez et al. / IFAC PapersOnLine 53-2 (2020) 7635–7640 7639 ∂tˆum l=∂rrn−1∂rˆum l rn−1−l(l+n−2)  r2ˆum l+λ(r)ˆum l +pn lm(r)(∂rum l(t, R)−∂rˆum l(t, R)),(23) with boundary condition ˆ Um l(t)=Um l(t).(24) We need to design the output injection gain pn lm(r). Closely following [20], define the observer error as ˜u=u−ˆu. The observer error dynamics are given by ∂˜ulmt ∂t = rn−1∂rrn−1∂r˜ulm−l(l+n−2)  r2˜ulm +λ(r)˜ulm −pn lm(r)∂r˜ulm(t, R),(25) with boundary conditions ˜ulm(t, R)=0.(26) Next we use the backstepping method to find a value of pn lm(r)that guarantees convergence of ˜uto zero. This ensures that the observer estimates tend to the true state values. Our approach to design p(r)is to seek a mapping that transforms (25) into the following target system ∂˜wlmt ∂t = rn−1∂rrn−1∂r˜wlm −c˜wlm −l(l+n−2)  r2˜wlm,(27) with boundary conditions ˜wlm(t, R)=0.(28) The transformation is defined as follows: ˜ulm(t, r)= ˜wlm(t, r)−R r Pn lm(r, ρ)˜wlm(t, ρ)dρ, (29) and then pn lm(r)will be found from transformation kernel as an additional condition. From [20], one obtains the following PDE that the kernel must verify: 1 rn−1∂rrn−1∂rPn lm−∂ρρn−1∂ρPn lm ρn−1 −l(l+n−2) 1 r2−1 ρ2Pn lm =−λ(r) Pn lm (30) In addition we find a value for the output injection gain kernel pn lm(r)=Pn lm(r, R)(31) Also, the following boundary condition has to be verified 0=λ(r)+(∂rPn lm(r, ρ)) ρ=r + rn−1 d dr rn−1Pn lm(r, r) +∂ρPn lm(r, ρ) ρn−1ρ=r rn−1,(32) which can be written as 0=λ(r)+∂rPn lm(r, r)+d dr (Pn lm(r, r))+(n−1)Pn lm(r, r) r +∂ρPn lm(r, r)−(n−1)P n lm(r, r) r.(33) Following [20], and after some computations, we reach boundary conditions for the kernel equations as follows Pn lm(0,ρ)=0,∀l=0 (34) Pn lmr(0,ρ)=0,∀l=1 (35) Pn lm(r, r)=−r 0λ(σ)dσ 2.(36) It turns out the observer kernel equation can be transformed into the control kernel equation, therefore obtaining a similar explicit result. For this, define ˇ Pn lm(r, ρ)=ρn−1 rn−1Pn lm(ρ, r),(37) and it can be verified that the equation now verified by ˇ Pn lm(r, ρ)is exactly the equation verified by Kn lm(r, ρ). Thus ˇ Pn lm(r, ρ)=Kn lm(r, ρ)and we can apply our previous result of Section 4. The observer error ˜uhas the same stability properties derived in Section 3 for the closed-loop system under the full-state control. As in the controller case, only a limited number of modes need to be estimated; namely, those that are not naturally stable by virtue of Lemma 2, this being the main difference with the result given in [20]. Finally, the controller-observer augmented system can be proved closed-loop stable as in [20], using the separation principle given the linearity of the system, with desired convergence rate, and without much modification; we skip the details for lack of space, which requires going up to H1 stability, as in [20]. 6. SIMULATION STUDY In this section, the simulation experiment on threedimensional ball (n=3) is taken as an example to illustrate the effectiveness of proposed control. The system with the output-feedback control law is simulated over 0≤t≤2s with the following parameters: =1,λ(r) = 10r4+50r2+ 50,c=3. We consider that the system is initially at the random quantity, u0∈[0,10], and the observer’s initial condition is set as actual state plus an error of normal distribution with zero mean and σ2=0.5. Fig. 1 shows the plots of the polynomial approximation of kernels K3 lm, which is obtained by using (22) up to a cutoff at the p-th powers. The value of Kdoes not depend on mso we omit that sub-index. The value of pis chosen as p= 15. Applying Lemma 1, one can obtain Lto be 11; however, here to save space, we only show the first six approximate numerical solutions of control gains. As shown in Fig. 1, we find that the Klbecomes increasingly smaller when lincreases. In order to avoid a dramatic increase in the complexity of simulation caused by the high dimension, in our simulations we employ a method also based on spherical harmonics expansions which greatly reduces the error. Thus, we only calculate the harmonics ulm which only need discretization in the radial direction, and then sum a finite number Sof harmonics to recover u. When S>0 is a large enough integer, the error caused by the use of a finite number of harmonics is much smaller than the angular discretization error. Thus, the simulation is carried out using the formula u(t, r, θ1,θ 2)= l=S  l=0 m=l  m=−l um l(t, r)Y3 lm(θ1,θ 2)(38) where the spherical harmonics are defined as -250 1 -200 -150 1 K0(r, ) -100 0.8 -50 0.5 0.6 r 0 0.4 0.2 00 -120 1 -100 -80 1 -60 K1(r, ) -40 0.8 0.5 -20 0.6 r 0 0.4 0.2 00 -80 1 -60 1 -40 K2(r, ) 0.8 -20 0.5 0.6 r 0 0.4 0.2 00 (a) l=0 (b) l=1 (c) l=2 -50 1 -40 -30 1 K3(r, ) -20 0.8 -10 0.5 0.6 r 0 0.4 0.2 00 -40 1 -30 1 -20 K4(r, ) 0.8 -10 0.5 0.6 r 0 0.4 0.2 00 -40 1 -30 1 -20 K5(r, ) 0.8 -10 0.5 0.6 r 0 0.4 0.2 00 (d) l=3 (e) l=4 (f) l=5 Fig. 1. Polynomial approximation of the first six control gains Kl(r, ρ),l=0,1,...,5. (a) t=0s (b) t=0.18s 0 0.5 1 1.5 2 time(s) 0 2 4 6 8 10 average value of u(t,r, 1,2) 1049 (c) t=0.2s (d) Mean of the open-loop system Fig. 2. Open-loop evolution. Y3 lm(θ1,θ 2)=(−1)m2l+1 4π (l−m)! (l+m)!P3 lm(cos(θ1))ejmθ2 (39) with P3 lm the associated Legendre polynomial defined as P3 lm(s)= 1 2ll!(1 −s2)m/2dl+m dsl+m(s2−1)l(40) Fig. 2 and Fig. 3 illustrate the transients of open-loop and closed-loop responses at different times, respectively, where the colour denotes the value of the position at this time. The evolutions of average value of uare plotted in Fig. 2(d) and Fig. 3(e), respectively. (Note that the ranges of color bars are different. And thus avoid the appearance of almost similar colors in fig. 3 of using same upper limit.) Comparing the open-loop and closed-loop evolution, the validity of proposed method is illustrated more intuitively. Fig. 3(f) shows the average of the observation errors, from which it can be found that the system begins to converge to its zero equilibrium after the observation error has already settled to zero as well. The evolutions at different layers, namely r=0.002,r=0.3,r=0.5and r=0.8, are shown in Fig. 4 (a), (c) , as well as the observer errors are presented in Fig. 4 (b), (d). For clarity, only the first 0.4s of response are shown here. Fig. 5 depicts control effort at (a) t=0.1s (b) t=0.2s (c) t=0.4s (d) t=2s 0 0.5 1 1.5 2 time(s) 0 1 2 3 4 0 0.5 1 1.5 2 time(s) 0 0.5 1 1.5 2 2.5 3 (e) Mean (closed-loop system) (f) Mean (observer error ˜u) Fig. 3. Closed-loop evolution using output feedback control. (a)-(d) Transient states. Note that the different upper ranges of the color bars in Fig. 2 and 3. (e) Mean of system state u. (f) Mean of observer error ˜u. 0 0.1 0.2 0.3 0.4 time(s) 0 10 20 30 40 50 60 u(t,r, ,0) r=0.02 r=0.3 r=0.5 r=0.8 0 0.1 0.2 0.3 0.4 time(s) 0 2 4 6 8 10 12 r=0.02 r=0.3 r=0.5 r=0.8 (a) Actual states at (r, π, 0) (b) Observer errors at (r, π, 0) 0 0.1 0.2 0.3 0.4 time(s) 0 10 20 30 40 50 u(t,r, , /4) r=0.02 r=0.3 r=0.5 r=0.8 0 0.1 0.2 0.3 0.4 time(s) 0 2 4 6 8 10 r=0.02 r=0.3 r=0.5 r=0.8 (c) Actual states at (r, π, π/4) (d) Observer errors at (r,π,π/4) Fig. 4. The details of closed-loop evolution at different r or θ. (a) (c) Actual states. (b) (d) Observer errors between the actual and estimated states. the boundary. It can be seen that the system driven by the proposed boundary control eventually converges after a short-term fluctuation. 7640 Rafael Vazquez et al. / IFAC PapersOnLine 53-2 (2020) 7635–7640 0 0.1 0.2 0.3 0.4 time(s) 0 20 40 60 80 100 120 U(t, 1,/4) 1 =0 1= /4 1= 1=3 /2 0 0.025 0.05 0.075 0.1 0 0.1 0.2 0.3 0.4 time(s) 0 20 40 60 80 100 U(t, 1,3 /8) 0 0.025 0.05 0.075 0.1 =0 1=/4 1= 1=3 /2 (a) Control effort at (θ1, π/4) (b) Control effort at (θ1,3π/8) Fig. 5. The control effort at different θ. 7. CONCLUSION We have shown a design to stabilize a radially-varying reaction-diffusion equation on an n-ball, by using an output-feedback boundary control law (with boundary measurements as well) designed through a backstepping method. The radially-varying case proves to be a challenge as the kernel equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations and successive approximations become difficult to use. Using a power series approach, a solution is found, thus providing a numerical method that can be readily applied, to both control and observer boundary design. In addition the required conditions for the radially-varying coefficients are revealed (analyticity and evenness). In practice, this result can be of interest for deployment of multi-agent systems, by following the spirit of [13]; thus, the radial domain mirrors a radial topology of interconnected agents which follow the reaction-diffusion dynamics to converge to equilibria, that represent different deployment profiles. Since one can choose the plant as desired (thus setting the behaviour of the agents), using analytic reaction coefficients is not actually a restriction, but rather opens the door to richer families of deployment profiles compared with the constant-coefficient case of [13]. On the other hand, the theoretical side of the result needs to be further investigated; besides convergence issues, another avenue of research that can be explored is the relaxation of the analyticity hypothesis by using reaction coefficients belonging to the Gevrey family; the kernels can then be analyzed to verify if they are still analytic, or rather Gevrey-type kernels, or simply do not diverge. REFERENCES [1] G. Andrade, R. Vazquez and D. Pagano, “Backstepping stabilization of a linearized ODE-PDE Rijke tube model,” Automatica, vol. 96, 98-109, 2018. [2] P. Ascencio, A. Astolfi and T. Parisini, “Backstepping PDE design: A convex optimization approach,” IEEE Transactions on Automatic Control, vol. 63, pp. 1943–1958, 2018. [3] J. Auriol and F. Di Meglio, “Minimum time control of heterodirectional linear coupled hyperbolic PDEs,” Automatica, vol. 71, pp. 300–307, 2016. [4] K.Atkinson and W. 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