Necessary conditions of optimality for impulsive control without a priori normality assumptions

journal of optimization theory and applications: Vol. 124, No. 1, pp. 55–77, January 2005 (© 2005) Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions1 A. Arutyunov,2V. Dykhta,3and F. Lobo Pereira4 Communicated by B. Polyak Abstract. First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result. Key Words. Optimal impulsive control, extremal principle, secondorder optimality conditions, abnormality. 1. Introduction Let us consider the following fixed-time optimal control problem: (A) min J(x 0,u,w)=L0(a), (1) s.t. dx(t) =f (t, x(t), u(t))dt +G(t, x(t))dw(t), t ∈[t0,t 1],(2) 1The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant. 2Professor, Department of Differential Equations and Functional Analysis, Peoples Friendship University of Russia, Moscow, Russia. 3Professor, Department of Mathematics, Baikal State University of Economy and Law, Irkutsk, Russia. 4Associate Professor, Department of Electrotechnical and Computer Engineering, Faculty of Engineering, University of Porto, Porto, Portugal. 55 0022-3239/05/0100-0055/0 © 2005 Springer Science+Business Media, Inc. 56 JOTA: VOL. 124, NO. 1, JANUARY 2005 L1(a) ≤0,L 2(a) =0,(3) dw∈K.(4) Here, a=(x(t0), x(t1)), x(t0)=x(t− 0)=x0,x(t 1)=x1,t 0<t 1 are given. The mappings f:[t0,t 1]×Rn×Rm→Rn,G:[t0,t 1]×Rn→Rn×k, Li:Rn×Rn→Rd(Li),i=0,1,2, are given, with d(Li)the dimension of the vector function Li,d(L 0)=1, and dw is a k-dimensional Borel measure associated with the function of bounded variation w(t), right continuous on (t0,t 1].The cone Kis defined by K={dw∈C∗([t0,t 1];Rk):∀continuous φsuch that φ(t)∈K0∀t, B φ(t)dw≥0,∀Borel B⊂[t0,t 1]}, where Kis a given convex, closed, pointed cone from Rkand K0is its dual. In another words, the measure dw satisfies B dw(t) ∈K, for all Borel subsets B. The pair (u, w) is called an admissible control if u∈Lm ∞and w∈BVkis such that dw∈K. Let us describe our assumptions for problem A: (H1) The functions L0,L 1,L 2are C2. (H2) The function fis twice differentiable w.r.t. xand ufor almost all t∈[t0,t 1]; the function fplus the first-order and secondorder derivatives are measurable w.r.t. tand bounded on any bounded subset. (H3) The matrix function G∈C2. (H4) The matrix Gsatisfies the Frobenius condition, i.e., Gi x(t, x)Gj(t, x) −Gj x(t, x)Gi(t, x) ≡0,(5) where Giis the ith column of G. JOTA: VOL. 124, NO. 1, JANUARY 2005 57 Notice that, under (H4), the dynamic system (2) is robust w.r.t. approximations of the generalized control dw by conventional controls v(·)∈ Lk ∞([t0,t 1];K); see Refs. 1–4. If the Frobenius condition holds, then for any given admissible control (u, w) and initial condition x0, the corresponding trajectory (whose existence is assumed) is the unique right-continuous function of bounded variation on (t0,t 1],with x(t0)=x0such that x(t)=x0+t t0 f (θ, x(θ), u(θ))dθ +[t0,t] G(θ, x(θ))dwc(θ) + Si≤t (z(1;si,ci)−x(s− i)). (6) Here, dwcrepresents the continuous part of dw, dwa(t) :=ciδSi is the atomic part, si∈[t0,t 1] are the jump times of dw (times of impulses), δSis the Dirac measure at time s,ci∈Kare the jumps of dw, and the function zi(τ) =z(τ;si,c i)is the solution to the limiting system dzi/dτ =G(si,z i)ci,z i(0)=x(S− i);(7) hence, zi(1)=x(s+ i). The robustness of the system (2), due to (H4), implies that the solution (6) belongs to the closure of the set of absolutely continuous solutions of equation (2) corresponding to (u, w) ∈L∞×AC. An admissible control process is a triplet (x0,u,w), where (u, w) is an admissible control and the corresponding state trajectory satisfies the given endpoint constraints. The problem under consideration is to minimize J over the set of admissible control processes. By (x∗ 0,u ∗,w∗)and x∗, we denote respectively an admissible control process and the corresponding state trajectory investigated for a minimum of problem (A). It is assumed that this control process satisfies the following additional assumption: (H5) dw∗(t) =v∗(t)dt + s∈S∗ csδs(t), (8) 58 JOTA: VOL. 124, NO. 1, JANUARY 2005 where ν∗(t) =˙w∗(t) a.e. with respect to the Lebesgue measure on [t0,t 1]5, S∗⊂[t0,t 1] is the set of jump times of w∗(·), assumed to be finite, and cs=[w∗(s)]:=w∗(s+)−w∗(s−), i.e., the function w∗(·)has no singular continuous part and has a finite number of jump times. Moreover, since (x∗ 0,u ∗,w∗)is investigated for a local minimum only (in the sense of Definition 1.1 below), then without loss of generality we can assume that all endpoint inequality constraints are active at the optimal trajectory x∗,i.e., L1(a∗)=0,where a∗=(x∗(t0), x∗(t1)). (9) Dynamic optimization problems arising in a variety of application areas such as finance, mechanics, resources management, and space navigation (see Refs. 4–10, just to mention a small but representative sample of references), whose solutions might involve discontinuous trajectories, have been considered over the years, motivating a significant research effort on the impulsive control problem. In order to not obscure the aim of this article, we selected the simplest control problem paradigm enabling us to deal with the issues relevant to first-order and second-order conditions for impulsive control problems that remain informative, even for abnormal control processes. It is not difficult to see that this result can be derived for a number of different and more complex control formulations. In particular, by standard statevariable manipulations, one can convert Bolza and Lagrange types of cost functionals into the one stated here. The approach of this article can be used to derive these optimality conditions for problems with regular control constriants of the type R(u, t) =0. Under regularity assumptions (see Ref. 11), the implicit function theorem can be used to solve (for each t) this equation in u, thus converting the control problem into the one considered here. Definition 1.1. We say that the admissible process (x∗ 0,u ∗,w∗)is a local minimizer of the problem (A) if ∃ε>0 and, for any finite-dimensional subspace R⊂Lm ∞[t0,t 1],∃εR>0 such that process (x∗ 0,u ∗,w∗)yields the minimum to problem (1)–(4) with the additional constraints a−a∗<ε, dw−dw∗C∗([t0,t1];Rk)<ε, u−u∗Lm ∞[t0,t1]<ε R,u(·)∈R. 5Heretofore, L-a.e. denotes a.e. w.r.t. the Lebesgue measure. JOTA: VOL. 124, NO. 1, JANUARY 2005 59 The defined type of local minimum is finite dimensional in uand weak in dw. In this article, we obtain first-order and second-order necessary conditions of optimality for the problem under consideration. The main features of the results is that no a priori normality assumptions are required and that they are informative for abnormal control processes as well. Another issue concerns the fact that, in the problem considered, the function G depends also on x. The proof of these conditions is based on a nonlinear transformation of the initial problem A (Ref. 4) into another one for which Gdoes not depend on xand first-order and second-order necessary conditions of optimality were derived in Ref. 12. In spite of the well-developed theory of higher-order necessary conditions of optimality for conventional optimal control problems (see for example, Refs. 11, 12, 14), it is somewhat surprising that, from the vast amount of literature addressing optimal impulsive control problems (Refs. 1–3, 15–22), only a few publications are available (Refs. 23–25, 27). We notice that, while the conditions in Refs. 23, 24 become trivial (i.e., degenerate, for abnormal problems), ours remain informative. Also, our results differ substantially from these conditions as it can be seen from the fact that these follow directly from the maximum principle in the case the optimal trajectory is absolutely continuous, i.e., with no impulses. In Ref. 3, second-order necessary conditions of optimality of the Legendre-Jacobi-Morse type for time-optimal control are derived by using in an essential way an extremal principle and the notion of index of quasiextremality provided in Ref. 26. However, the approach followed here differs substantially from all the ones in the references cited above as we regard this problem as a specific instance of a general abstract problem for which powerful second-order optimality conditions are derived. This article is organized as follows. In Section 2, we introduce key definitions and state first-order and second-order necessary conditions of optimality for the dynamic optimization problem described in Section 1. Issues concerning abnormality, geometric interpretation, and computation are also discussed. In Section 3, we present the proof, which is organized in three parts: transformation of the given problem into another one for which there are first-order and second-order necessary conditions of optimality available; statement of the mentioned optimality conditions for the problem considered; and decoding of the thus obtained first-order and of second-order conditions in terms of the data of the original problem. Finally, in Section 4, two examples illustrate the application of these conditions. 60 JOTA: VOL. 124, NO. 1, JANUARY 2005 2. Second-Order Necessary Conditions of Optimality Let us state the necessary conditions of optimality for problem A. Before presenting the main result, we discuss some auxiliary concepts which are fundamental for the statement of our main result: local maximum principle, critical cone, and quadratic form. 2.1. Local Maximum Principle. Let F(t,x,u,v)=f(t,x,u)+G(t, x)v and let ψ∈Rn,λ=(λ0,λ 1,λ 2)∈R1×Rd(L1)×Rd(L2). Define the Pontryagin function H=H0+H1and the endpoint Lagrangian lλby H0(t,x,ψ,u)=ψ,f (t, x, u), H1(t,x,ψ,v)=ψ, G(t, x)v, lλ(a) =λ0L0(a) +λ1,L 1(a)+λ2,L 2(a). Definition 2.1. We say that a process (x∗ 0,u ∗,w∗)satisfies the EulerLagrange conditions or the local maximum principle if there exists λ=0 such that λ0≥0,λ 1≥0,λ1,L 1(a∗)=0 (10) and the vector function ψ, solution to the adjoint system −dψ(t) =H0x(t)dt +Hxv(t)dw∗(t), −ψ(t1)=lλ x1(a∗), (11) which satisfy the following conditions: ψ(t0)=lλ x0(a∗), (12) Hu(t) =0,L-a.e.,(13) Hv(t), v≤0,∀(t, v) ∈[t0,t 1]×K, (14) Hv(t), ¯ω∗(t)=0,dw ∗-a.e.,(15) where ¯ω∗(t) =dw∗(t)/d|w∗(t)| JOTA: VOL. 124, NO. 1, JANUARY 2005 61 is the Radon-Nicodym derivative of the measure dw∗with respect to its total variation measure. Notice that the solution to the adjoint system (11) is in the same sense as the one to (6), i.e., ψ(t1)=−lλ x1(a∗) and that ψ(t)=−lλ x1(a∗)+t1 t H0x(θ)dθ +t1 t Hxv(θ)dw∗ c(θ) + si>t (ψ(si)−q(0;si,ci)), t ∈[t0,t 1). (16) Here, the functions qi(τ)=qi(τ ;si,ci)are solutions to the adjoint limiting system −dqi/dτ =Hxv(si,z i(τ), qi(τ))ci,q i(1)=ψ(si), (17) with the corresponding solution zi(τ) to the system (7) when x(s− i)= x∗(s− i). The notation Hxv(t) =∂2H ∂v∂x(t) refers to the evaluation of the function Hxv along the process examined (this notation is adopted also for other functions in similar contexts). We remark that any adjoint trajectory ψ(t) and the function H(t) depend on λdue to the transversality condition (12). Denote by =(x∗ 0,u ∗,w∗) the set of all normalized Lagrange multipliers λ,λ=1, satisfying the local maximum principle. It is well known that =∅ is a first-order necessary condition for a weak local minimum for problem A. However, we shall prove here that it is also necessary for the local minimum in the sense of Definition 1.1. Note that the local maximum principle holds without (H5). 62 JOTA: VOL. 124, NO. 1, JANUARY 2005 2.2. Critical Cone. In order to ensure a compact statement of the second-order conditions, we shall use the total derivative w.r.t. time along the solution to the following ordinary differential system: ˙x=F (t, x, u, v), (18a) −˙ ψ=Hx(t,x,ψ,u,v), (18b) ˙w=v, v(t)∈K. (18c) For example, (˙ Hv)x=(∂/∂x)[(d/dt)(∂H/∂v)]|t,x∗(t),u∗(t),w∗(t)). Under the Frobenius condition, this derivative does not depend on v,but in any other case, we put always v∗(t)=˙w∗(t); see (8). Denote by BVn(S∗) the set of n-dimensional vector functions of bounded variation whose jump times are supported on S∗. Clearly, each term in (x∗(·), ψ(·), w∗(·)) isinaBV(S∗)space of the corresponding dimension. Definition 2.2. A variation (δx0, δu, δw) ∈Rn×Lm ∞×BVk(S∗)is called critical if the corresponding state trajectory variation δx ∈BVn(S∗) satisfies the following conditions: Lia(a∗), δa+Lix1(a∗), G(t1)δw1≤0,i=0,1, =0,i=2,(19) δa =(δx(t0), δx(t1)), δw1=δw(t1), (20) d(δx)/dt =Fx(t)δx +Fu(t)δu −(˙ Hv)T ψ(t)δw, t /∈S∗,(21) d(δw)∈K+Lin {dw∗},δw(t 0)=0,(22) δx(s) =δq(1;s,c), ∀s∈S∗.(23) Here, G(t1)=G(t1,x∗(t1)), δq(τ;s,c):=δqs(τ ) is the solution to the system d(δqs)/dτ =H1ψx(s, zs(τ), c)δqs,(24a) δqt0(0)=δx0,(24b) δqs(0)=δx(s−), s > t0,(24c) and the function zs(τ) is solution of (7) when si=s,x(s−)=x∗(s−); recall that c=[w∗(s)]. Denote by Kcr the cone of all critical variations. JOTA: VOL. 124, NO. 1, JANUARY 2005 63 2.3. Quadratic Form. For any λ∈, define the quadratic form λ(δx0, δu, δw) =δaTlλ aa(a∗)δa +Qλ 1(δa, δw1) −t1 t0 Qλ(δx, δu, δw)(t)dt, (25) where Qλand Qλ 1are the following quadratic forms: Qλ(δx, δu, δw) =δuTHλ uuδu+2δxTHλ xuδu−2δwT(˙ Hλ v)uδu −δwT(¨ Hλ v)vδw−2δwT(˙ Hλ v)xδx +δxTHλ xxδx, (26) Qλ 1(δx(·), δw1)=2[δx(t0)Tlλ x0x1(a∗)G(t1)−δx(t1)THλ xv(t1)]δw1 +δwT 1GT(t1)[lλ x1x1(a∗)G(t1)−Hλ xv(t1)]δw1 − s∈S∗ [δxT(s)λ(s)δx(s) −δxT(s−)λ(s−)δx(s−)]. (27) Here, dependence on top Qλis omitted, Hλrefers to the Pontryagin function evaluated along (t, x∗,ψ,u ∗,v∗), ψ satisfied (11) for a certain λ, and δx(·)is the corresponding solution to (24) with (22), (23), δx(t− 0)= δx0,w(t− 0)=0. Only λ(t) ∈BVn×n(S∗)remains to be defined in formula (27). For this, let z∗(τ;t) and q∗(τ;t) be solutions of dz∗/dτ =G(t, z∗)w∗(t), z∗(1;t)=x∗(t), −dq∗/dτ =H1x(t, z∗,q∗,w∗(t)), q∗(1;t)=ψ(t), and let the n×nmatrix Z(τ;t) satisfy −dZ/dτ =ZH1ψx(t, z∗(τ, t), w∗(t)), Z(0;t)=I. Then, λ(t) =−ZT(1;t)1 0 Z−1T(τ;t)H1xx(τ;t)Z−1(τ ;t)dτZ(1;t), (28) where the expression H1xx(τ;t) is a short notation for H1xx evaluated along (t, z∗(τ;t),q∗(τ;t),w∗(t)). We remark that (t− 0)=0 because w(t− 0)=0. 70 JOTA: VOL. 124, NO. 1, JANUARY 2005 where δy(·)is the corresponding solution to (47). Put ˜ Kπ=Ker ˜ A. Proposition 3.2. We have that ˜ d=codim (Im ˜ A)and also µ∈Mais equivalent to the fact that the index of the form ˜ µ aconsidered on the linear subspace ˜ Kπis not greater than ˜ d. This proposition follows from the fact that BVkis dense in Lk ∞with respect to the Lk 2metric and its proof is based on standard Lebesgue integration arguments. Step 3. Decoding of the Local Maximum Principle. From the definition of ξand η, it follows that ηsatisfies the partial differential system ηw(t,x,w)+ηx(t, x, w)G(t, x) =0,(52) with boundary condition η(t,x,0)=x. Moreover, the following equalities hold: η(t,ξ(t,y,w),w)=y, ξ(t,η(t,x,w),w)=x. (53) We shall use these relations and their consequences obtained by differentiation. Let us consider the following proposition. Proposition 3.3. The function σ(t,y,p,w) =ηT x(t, ξ(t, y, w), w)p is a solution to the following completely integrable system: σw=−Hxv(t, ξ(t, y, w), σ, u, v), σ |w=0=p. (54) Proof. Obviously, σsatisfies the initial condition. Denote the lefthand side of (52) by F(t,x,w). By differentiating σwith respect to w, and using Fx≡0, we obtain σw(t,y,p,w)=ηT xx(t, ξ(t, y, w), w)GT(t, ξ(t, y, w))p +ηwx(t, ξ(t, y, w), w)p =Fx(t, ξ(t, y, w), w)p −GT x(t, ξ(t, y, w))ηT x(t, ξ(t, y, w), w)p =−Hxv(t, ξ(t, y, w), σ (t, y, p, w)). This proves Proposition 3.3. JOTA: VOL. 124, NO. 1, JANUARY 2005 71 Proposition 3.4. Let s(t,x,ψ,w)=ξT y(t, ηx(t,x,w),w)ψ. Then, the function p(t)=s(t,x∗(t), ψ(t), w∗(t)) (55) is a solution to (36) if and only if the function ψ(t)=σ(t,y∗(t), p(t), w∗(t)) (56) is solution to the system (11). Proof. Any solution (x(·), ψ(·)) to the systems (2), (11) can be approximated in the weak star topology of the space of functions of bounded variation by solutions to the system (18). Hence, it is sufficient to prove that the relations (55) and (56) hold for any absolutely continuous trajectory of the system (18). For this, it suffices to prove that the formula (d/dt)σ (t, y, p, w) =−Hx(t,ξ(t,y,w),σ(t,y,w),u,v) (57) holds, with (˙y, ˙p, ˙w)=(g, −˜ Hy,v). We have (d/dt)σ (t, y, p, w) =−ηT x(t,ξ(t,y,w),w) ˜ Hy(t,y,p,u,w) +(d/dt)ηT x(t,ξ(t,y,w),w)p. (58) Note that the function ˜ Hmay be represented in the form ˜ H(t,y,p,u,w)=H0(t,ξ(t,y,w),σ(t,y,p,w),u) +p,ηt(t,ξ(t,y,w),w). By differentiating w.r.t. y, we obtain ˜ Hy(t,y,p,u,w)=ξT y(t,y,w)[H0x(t,ξ(t,y,w),σ(t,y,p,w),u) +[ηT x(t, ξ(t, y, w), w)p]xf(t,ξ(t,y,w),u) +ηtx(t, ξ(t, y, w), w)p]. 72 JOTA: VOL. 124, NO. 1, JANUARY 2005 By substituting this equality in (58), we obtain (d/dt)σ (t, y, p, w) =−H0x(t,ξ(t,y,w),σ(t,y,p,w),u) −[ηT x(t, ξ(t, y, w), w)p]xf(t,ξ(t,y,w),u) −ηT tx(t, ξ(t, y, w), w)p +(d/dt)ηT x(t,ξ(t,y,w),w)p. (59) By considering the identity Fx(t,x,w)≡0 and using it in the last term of (59), we obtain (d/dt)[ηT x(t, ξ(t, y, w), w)p]x =[ηT x(t, ξ(t, y, w), w)p]t+[ηT x(t, ξ(t, y, w), w)p]xf(t,ξ(t,y,w),u) −Hxv(t,ξ(t,y,w),ζ(t,y,p,w),u,v)v. By substituting in (59), we obtain (57). This proves Proposition 3.4. In a similar way (see the details in Refs. 4, 18, 23), we can obtain the important equality −˙pw=˜ Hw(t,y,p,u,w)=−(d/dt)Hv(t, ξ(t, y, w), σ (t, y, p, w)). (60) Let us consider the transversality conditions for the adjoint systems. We have p(t0)=˜ Ly0(b∗)=ξT y(t0,y∗(t0), 0)Lx0(a∗)=Lx0(a∗), −p(t1)=˜ Ly1(b∗)=ξT y(t1,y∗(t1), w∗)Lx1(a∗), −pw(t1)=˜ Lw1(b∗)=GT(t1,x∗(t1))Lx1(a∗)=Hv(t1,x∗ 1,ψ(t 1)). From these relations and Proposition 3.4, it follows that the set of adjoint trajectories p(·)and ψ(·)can be obtained from each other by coordinates transformation. Moreover, bearing in mind the equality (60), we obtain pw(t) =−Hv(t), ∀t∈[t0,t 1], since under (H4) the function t→Hv(t) ∈AC. Due to this and the obvious equality ˜ Hu(t) =Hu(t), we conclude that the sets and Mare distinguished only by the notation. From now on, we shall use the common notation λand instead of µand M. This conclusion is stated as follows. JOTA: VOL. 124, NO. 1, JANUARY 2005 73 Proposition 3.5. =∅⇔M=∅. That is, the conditions of the local maximum principle in problems A and B are equivalent. Step 4. Decoding of the Second-Order Conditions. We shall prove that the cone ˜ Kcr and the form ˜ λare transformed into Kcr and λ respectively by the simple linear mapping of the y-variations :δx(t) =ξy(t)δy(t). (61) Since this transformation is invertible [det ξy(t)=0on[t0,t 1] due to (33)], then in order to obtain the second-order necessary conditions of optimality for problem A from those for problem B, we need to show the following proposition. Proposition 3.6. The following equalities hold: ◦˜ Kcr =Kcr, ◦˜ λ=λ,∀λ∈. (62) Proof. The linear relations (44), (45) specifying the critical cone can be transformed easily into (19). Let us prove that, for t/∈S∗, the following formulas hold: gy(t) =(ηx(t)fx(t) +˙ηx(t))ξy(t), (63) gw(t) =ηx(t)(fx(t)G(t) −˙ G(t)) =−ηx(t)( ˙ Hv)ψ(t), (64) gu(t) =ηx(t)fu(t). (65) In fact, by using the equality Fx≡0, we obtain gy(t) =[ηxx(t)f (t) +fx(t)ηx(t) +ηtx(t)]ξy(t) ={[Fx(t)ηx(t) +(d/dt)ηx(t)]−Fx(t)v∗(t)}ξy(t). Analogously, by using the additional equality Ft≡0, we obtain (64) and (65). From (63)–(65) and (61), we see that (43a) is rewritten in the form (21). For any s∈Sd(w∗), let z(τ;s)=ξ(s,y∗(s), w∗(s−)+τ[w∗(s)]), τ ∈[0,1], −q(τ,s)=ξy(s, y∗(s), w∗(s−)+τ[w∗(s)])δy(s), τ ∈[0,1]. It is easy to check that these functions satisfy equations (7), (24) and describe the jump conditions of the variation δx in (23) and (24). The proof of the second equality in (64) is analogous to the corresponding one in Ref. 23; therefore, it is omitted. 74 JOTA: VOL. 124, NO. 1, JANUARY 2005 From Propositions 3.2 and 3.6, it follows that condition (31) is equivalent to condition (50) for problem (B); hence, (31) is obtained. Theorem 3.1 is proved. 4. Examples Example 4.1. Take n≥5,k=n−1,x=col(x1,... ,x n)∈Rn, and let Q be a symmetric k×kmatrix such that the index of each of the matrices Q and −Qis not less than 2. The case k=4 and Q=diag(1,1,−1,−1)is a good example. Consider the problem min J=ζ,(x1(1),... ,x k(1)), s.t. dxi=fi(x, t)dt +dwi,i=1,k, w=col(w1,... ,w k), dxn=fn(x, t)dt +Qcol(x1,... ,x k), dw, t∈[0,1],x(0)=0,x n(1)=0,K=Rk, where ζ∈Rkis a given nonzero vector; for i=l,... ,n, let the fibe arbitrarily given smooth functions such that fi(0,t)≡0,f ix(0,t)≡0,f nxx(0,t)≡0. Because of the symmetry of Q, it can be shown easily that the Frobenius condition holds. We investigate the admissible control process (0,0,0) and prove that it is not a locally optimal control process. Fix any λ∈. From (14), for ψ(·)=ψλ(·)=(ψ1(·),...ψ n(·)),we obtain ψi(t) ≡0,i=l,... ,k, and from (11), we have ψn(t) ≡ψn,0=const. Hence, by using (11), (12), and ζ=0, we obtain ={λ:λ0=0,λ 2,i =0,i=1,n−1,λ 2,n =−λ2,n+1}; consequently, consists of only two vectors, ¯ ¯ λ=−¯ λ, ¯ λ=(1/√2)(0,... ,0,1,−1)and ψn,0=±1/√2. It can be shown easily that d=1, λ a(δw) =ψn,01 0Qcol(δx1,... ,δx k), δwdt. Hence, λ a(δw) =(1/2)ψn,0Qcol(δx1(1),... ,δx k(1)), col(δx1(1),... ,δx k(1)). JOTA: VOL. 124, NO. 1, JANUARY 2005 75 This implies that, for any ψn,0=±1, the index of the function λ ais not less than 2. So, a=∅; consequently, the process (0,0,0) is not optimal. Also notice that this process is abnormal, that max λ∈λ(δw) ≥0,∀δw, because of ¯ λ, ¯ ¯ λ∈, and that the last inequality is not useful. Example 4.2. Consider the following optimal control problem with parameters α1,α 2. min x3(1), s.t. dx1=x2dt, x1(0)=0, dx2=dw, x2(0)=x20 <0, dx3=(α1x1+α2x2)dw, x3(0)=0, dw≥0. The control function w∗(t) =−x20,∀t∈(0,1], with w∗(0)=0 satisfies the maximum principle for any parameter values. The corresponding trajectories and adjoint variables are, respectively, (x∗ 1(t), x∗ 2(t)) =(0,0), ∀t∈(0,1], with (x∗ 1(0), x∗ 2(0)) =(0,x 20), and (ψ1(t), ψ2(t) =(0,0), ∀t∈(0,1], with (ψ1(0), ψ2(0)) =x20(α1,α 2)and ψ3≡−1. The critical cone for the control is described by the conditions δ˙x1=δw, δx1(0)=δx1(0+)=0, δ˙x2=0,δx 2(0)=0, d(δw)∈C∗([0,1],R+)+γδ 0,γ∈R. 76 JOTA: VOL. 124, NO. 1, JANUARY 2005 Since H1ψx ≡0,δx 1(·), and δx2(·)are continuous; hence, δx2≡0. The fact that H1xx ≡0 implies that ψ≡0. Therefore, the form is given by (δw)=2α1δx1(1)δw1+α2δw2 1−2α11 0 δw(δw +δx1)dt. The necessary conditions of Theorem 2.3 amount to the inequality ≥0onkcr ; consequently, the function δw∗≡0 has to minimize  on Kcr. From the maximum principle, we have that α1≤0. Notice that it suffices to consider a needle-shaped variation δw concentrated at a left semineighborhood of point t=1. If α1<0 and α2>0, then the control w∗is globally optimal. References 1. Miller,B.,The Optimality Conditions in a Problem of Control of a System That Can Be Described with a Differential Equation with Measure, Avtomatika i Telemekhanika, Vol. 6, pp. 752– 761, 1982. 2. Zavalischin, S., and Sesekin, A., Impulsive Processes: Models and Applications, Nauka, Moscow, Russia, 1991. 3. Sarychev, A., Optimization of Generalized Controls in a Nonlinear Time-Optimal Problem, Differential Equations, Vol. 27, pp. 539–550, 1991. 4. Dykhta, V., and Samsonyuk,O.N.,Optimal Impulse Control with Applications, Nauka, Moscow, Russia, 2000. 5. Marec,J.,Optimal Space Trajectories, Elsevier, Amesterdam, Holland, 1979. 6. Lawden,D.,Optimal Trajectories for Space Navigation, Butterworth, London, England, 1663. 7. Brogliato,B.,Nonsmooth Impact Mechanics: Models, Dynamics, and Control, Lecture Notes in Control and Information Sciences, Springer Verlag, New York, NY, Vol. 220, 1996. 8. Clark,C.,Clarke, F., and Munro,G.,The Optimal Exploitation of Renewable Stocks, Econometrica, Vol. 47, pp. 25–47, 1979. 9. Krotov,V.,Bukreev, V., and Gurman,V.,New Variational Methods in Flight Dynamics, Mashinostroenie, Moscow, Russia, 1969. 10. Baumeister,J.,On Optimal Control of a Fishery, Proceedings of NOLCOS’01, 5th IFAC Symposium on Nonlinear Control Systems, St. Petersburg, Russia, 2001. 11. Arutyunov, A., Second-Order Necessary Conditions in Optimal Control Problems, Doklady Mathematics, Vol. 61, pp. 158–161, 2000. 12. Arutyunov, A., Optimality Conditions: Abnormal and Degenerate Problems, Kluwer Academic Publishers, Dordretch, Holland, 2000. 13. Krotov,V.,Global Methods in Optimal Control, Marcel Dekker, New York, NY, 1996. 14. Ledzewicz, U., and Schaettler, H., HigherOrder Conditions for Optimality, SIAM Journal on Control and Optimization, Vol. 37, pp. 33–53, 1998. JOTA: VOL. 124, NO. 1, JANUARY 2005 77 15. Bressan, A., and Rampazzo,F.,Impulsive Control Systems with Commutative Vector Fields, Journal of Optimization Theory and Applications, Vol. 71, pp. 67–83, 1991. 16. Miller, B., and Rubinovitch, E., Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic Publishers, Amsterdam, Holland, 2002. 17. Dykhta,V.,Conditions of Local Minimum for Singular Modes in the System with Linear Control, Automation and Remote Control, Vol. 12, pp. 5–10, 1981. 18. Dykhta,V.,Necessary Optimality Conditions for Impulsive Processes under Constraints on the Image of the Control Measure, Izvestiya Vyssikh Uchebnykh Zavedenyi Matematika, Vol. 12, pp. 1–9, 1996. 19. Kolokolnikova,G.,A Variational Maximum Principle for Discontinuous Trajectories of Unbounded Asymptotically Linear Control Systems, Journal of Differential Equations, Vol. 33, pp. 1633–1640, 1997. 20. Vinter, R., and Pereira,F.,A Maximum Principle for Optimal Processes with Discontinuous Trajectories, SIAM Journal on Control and Optimization, Vol. 26, pp. 205–229, 1988. 21. Pereira, F., and Silva,G.,Necessary Conditions of Optimality for VectorValued Impulsive Control Problems, Systems and Control Letters, Vol. 40, pp. 205–215, 2000. 22. Silva, G., and Vinter, R., Necessary Conditions for Optimal Impulsive Control Problems, SIAM Journal on Control and Optimization, Vol. 35, pp. 1829– 1846, 1997. 23. Dykhta,V.,Variational Maximum Principle and Quadratic Optimality Conditions for Impulsive and Singular Processes, Siberian Mathematical Journal, Vol. 35, pp. 70–82, 1994. 24. Dykhta,V.,Second-Order Necessary Optimality Conditions for Impulsive Control Problems and Multiprocesses, Singular Solutions and Perturbations in Control Systems, Edited by V. Gurman, B. Miller, M. Dmitriev, Pergamon, Elsevier Science, Kidlington, Oxford, UK, pp. 97–101, 1997. 25. Sumsonuk,O.,Quadratic Optimality Conditions for Optimal Impulsive Control Problems, Proceedings of the 12th Baikal International Conference on Optimization Methods and Their Applications, Irkutsk, Baikal, Russia, Vol. 2, pp. 144–149, 2001. 26. Agrachev, A., and Gamkrelidze, R., Index of Extremality and Quasiextremality, Russian Mathematics Doklady, Vol. 284, pp. 11–14, 1985. 27. Arutyunov, A., Jacimovic, V., and Pereira,F.,Second-Order Necessary Conditions for Optimal Impulsive Control Problems, Journal of Dynamical and Control Systems, Vol. 9, pp. 131–153, 2003.