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.
Pee e iew unde esponsibili y o In e na ional Fede a ion o Au oma ic Con ol.
10.1016/j.i acol.2024.09.028
10.1016/j.i acol.2024.09.028 2405-8963
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann ∗R. Hill ∗∗ A. Bempo ad ∗∗∗
G. Pannocchia ∗∗∗∗
∗Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗ Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann ∗R. Hill ∗∗ A. Bempo ad ∗∗∗
G. Pannocchia ∗∗∗∗
∗Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗ Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann ∗R. Hill ∗∗ A. Bempo ad ∗∗∗
G. Pannocchia ∗∗∗∗
∗Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗ Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann ∗R. Hill ∗∗ A. Bempo ad ∗∗∗
G. Pannocchia ∗∗∗∗
∗
Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗
Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann
∗
R. Hill
∗∗
A. Bempo ad
∗∗∗
G. Pannocchia ∗∗∗∗
∗Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗ Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Symbolic dynamics o ac i e se s o a class
o cons ained nonlinea op imal con ol
and MPC p oblems 1
M. M¨onnigmann ∗R. Hill ∗∗ A. Bempo ad ∗∗∗
G. Pannocchia ∗∗∗∗
∗Au oma ic Con ol and Sys ems Theo y, Ruh -Uni e si ¨a Bochum,
Ge many
∗∗ Elec ical and Elec onic Enginee ing, Uni e si y o Melbou ne,
Aus alia
∗∗∗ IMT School o Ad anced S udies Lucca, I aly
∗∗∗∗ Ci il and Indus ial Enginee ing, Uni e si y o Pisa, I aly
Abs ac : Solu ions o op imal con ol p oblems a e usually unde s ood o p o ide op imal
ajec o ies. In his pape , we show ha he op imal s a e-space sys em dynamics induce a
dynamics o he ac i e se s. Mo e speci ically, gi en he op imal ac i e se a he solu ion
ob ained a he cu en ime, i s successo op imal ac i e se (which, in u n, de ines he successo
solu ion) can be ound wi h index se ope a ions. These ope a ions do no in ol e any op imal
con ol (o o he op imiza ion o in eg a ion) p oblem, bu hey can be desc ibed wi h simple
ules. These ules cons i u e he symbolic dynamics o ac i e se s. The p esen pape ea s
a pa icula cons ained nonlinea p oblem class, ex ending ea lie esul s o he cons ained
linea -quad a ic case.
Keywo ds: op imal con ol, nonlinea model p edic i e con ol
1. INTRODUCTION
Cons ained op imal con ol p oblems a e mo e demand-
ing han hei uncons ained coun e pa s. Fo example,
disc e e- ime ini e-ho izon cons ained linea op imal con-
ol p oblems, such as hose ha a ise in linea model
p edic i e con ol (MPC) o mula ions, admi a simple
analy ical linea s a e- eedback solu ion in he absence o
cons ain s, while hey equi e mo e subs an ial online o
o line compu a ion e o in he p esence o linea con-
s ain s.
I has p o en o be use ul o cha ac e ize he solu ion o
cons ained op imal con ol p oblems wi h he se o ac i e
se s, whe e we call a subse o he indices o all cons ain s
ac i e se i his combina ion o cons ain s is ac i e o
a leas one easible ini ial s a e. The se o ac i e se s
hen is he se o all ac i e se s ha may appea when
sol ing he op imal con ol p oblem (OCP). Cons uc ing
he se o ac i e se s is o en associa ed wi h cons uc ing
explici solu ions (Gup a e al., 2011; He ceg e al., 2013;
Mi ze and M¨onnigmann, 2020; Obe dieck e al., 2017;
He ceg e al., 2015; Felle e al., 2013), speci ically o he
cons ained linea -quad a ic egula o (Bempo ad e al.,
2002; Tøndel e al., 2003; Se on e al., 2002).
E en i no explici eedback law is ul ima ely cons uc ed,
he se o ac i e se s o an OCP is s ill use ul, as i
de ines and cha ac e izes i s solu ion. We claim ha his
cha ac e iza ion is no only in e es ing pe se bu also
1MM and GP g a e ully acknowledge unding by he Eu opean
Commission unde g an no. 101079342 (F on Sea ).
in p ac ice. I an explici solu ion is no compu able,
ei he because i is oo expensi e o s o e and e alua e
(e en o mode a ely complex linea -quad a ic p oblems),
o no compu able a all (as in he gene al nonlinea case),
knowing he op imal ac i e se can g ea ly simpli y online
op imiza ion. Fo example in he simple case o uppe
and lowe bounds only, he OCP becomes uncons ained
and wi h a smalle o equal numbe o op imiza ion
a iables o de e mine. E en in case an explici solu ion is
p ede e mined o line, knowing he op imal ac i e se can
g ea ly simpli y he solu ion o he poin -loca ion p oblem
online.
The cha ac e iza ion wi h ac i e se s has led o se e al
in e es ing insigh s o he linea -quad a ic case. Neigh-
bo ing egions o he piecewise solu ion o he OCP can
be iden i ied om analyzing he ac i e se s (Ahmadi-
Moshkenani e al., 2018). Simila ly, i has been shown ha
dynamic p og amming can be ca ied ou wi h he ac i e
se s (M¨onnigmann, 2019), which helps o accele a e he
cons uc ion o all ac i e se s (Mi ze and M¨onnigmann,
2020). Finally, symme ies o he solu ion o he OCP can
be ound in he se o ac i e se s (Mi ze e al., 2023). All
hese esul s a e essen ially based on geome ic ela ions
o he egions de ined by he ac i e se s wi hou equi ing
geome ic calcula ions.
In he p esen pape , we ocus on dynamic ela ions
o ac i e se s o a pa icula nonlinea p oblem class.
Speci ically, we show ha he successo ac i e se can be
ound wi h e y simple ope a ions on he cu en ac i e
se (such as index shi s o index se s). In pa icula , no
Copy igh ©
2024 The Au ho s. This is an open access a icle unde he CC BY-NC-ND license
(
h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
)
182 M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187
op imal con ol p oblems (o o he op imiza ion p oblems)
need o be sol ed o ob ain successo ac i e se s. Successo
ac i e se s, which de ine he successo op imal solu ion,
can be cons uc ed bo h o open-loop op imal and closed-
loop op imal solu ions, whe e “closed-loop op imal” e e s
o he usual MPC use o he OCP solu ion on a eceding
ho izon. We no e ha dynamic ela ions o ac i e se s
ha e been used o de e mine obus MPC solu ions o
he linea -quad a ic case (M¨onnigmann and Pannocchia,
2020) and a sh inking ho izon nonlinea case (Dy ska and
M¨onnigmann, 2024) be o e.
Sec ion 2 in oduces he p oblem class. Sec ion 3 s a es
he main esul s. I i s in oduces he symbolic dynamics
in o mally (see (I), (II), (III) in subsec ion 3.3), illus a es
hese ideas wi h a sample p oblem, and s a es o mal
esul s. B ie conclusions and an ou look a e s a ed in
Sec ion 4.
2. PROBLEM STATEMENT
We conside disc e e- ime, nonlinea sys ems o he o m
x(k+ 1) = (x(k),u(k)),k=0,1,... (1a)
wi h s a e x(k)∈Rn, inpu u(k)∈Rm, and a nonlinea
s a e-upda e unc ion :Rn×Rm→Rn. We assume
s a es and inpu s a e subjec o cons ain s
x(k)∈X⊂Rn,
u(k)∈U⊂Rm(1b)
o all ime s eps k, whe e Xand Ua e compac se s
ha con ain he o igin in hei in e io and ha can be
desc ibed as he in e sec ion o a ini e numbe o suble el
se s. Fu he mo e, we assume is wice con inuously
di e en iable on an open supe se o X×U, and (0,0) =
0.
We a e in e es ed in he in ini e-ho izon nonlinea op imal
con ol p oblem o (1)
min
u(k),x(k+1),k=0,1,...
∞
k=0
ℓ(x(k),u(k)) (2a)
subjec o
x(k+ 1) = (x(k),u(k)),k=0,1,...
x(k)∈X,k=0,1,...
u(k)∈U,k=0,1,...
(2b)
o gi en cu en s a e x(0), and he s age cos ℓ(x, u),
which is speci ied la e . The ini e-ho izon p oblem o e
Ns eps
min
u(k),x(k+1),k=0,...N−1ℓT(x(N)) +
N−1
k=0
ℓ(x(k),u(k)) (3a)
subjec o
x(k+ 1) = (x(k),u(k)),k=0, ..., N −1
x(k)∈X,k=0, ..., N −1
u(k)∈U,k=0, ..., N −1
x(N)∈T.
(3b)
is used as an auxilia y p oblem, whe e ℓT(x) and T ⊆X
wi h 0 ∈in Ta e a e minal cos and cons ain , espec-
i ely, which a e desc ibed in mo e de ail in Assump ion 1.
Le FN e e o he se o ini ial s a es x(0) o which (3)
wi h ho izon Nhas a solu ion.
The ollowing Assump ion 1 allows o ela e solu ions
o (2) and (3). The assump ion is mild o linea sys ems
bu s ong o nonlinea sys ems. I he e o e dese es
some commen s, which a e s a ed below.
Assump ion 1. Assume he e exis a se T ⊆X,a
eedback law κ:T →Uand a e minal cos ℓT:T→R
such ha in Tis posi i e in a ian o he closed-loop
sys em and such ha , o any x(0) ∈T, e alua ing
u(k)=κ(x(k)),
x(k+ 1) = (x(k),κ(x(k))) (4)
o k=0,1,... yields a minimum o (2) o ini ial
condi ion x(0) wi h cos ∞
k=0 ℓ(x(k),u(k)) = ℓT(x(0)).
Assump ion 1 is s ong because i essen ially s a es we
know an op imal solu ion o he in ini e-ho izon p oblem
o all ini ial condi ions om Tand know he co e-
sponding in ini e-ho izon cos . This assump ion can eas-
ily be ul illed o linea -quad a ic p oblems unde mild
addi ional condi ions (Chmielewski and Manousiou hakis,
1996; Scokae and Rawlings, 1998), (Bempo ad e al.,
2002, Sec . 3). A nonlinea class ha espec s Assump-
ion 1 esul s o swi ching cos unc ions o he o m
ℓ(x, u)=˜
ℓ(x, u) i x∈T
ˆ
ℓ(x, u) o he wise.
This swi ching cos unc ion class yields he ollowing
ela ion o he in ini e-ho izon cos (2a) o he ini e-
ho izon cos (3a):
∞
k=0
ℓ(x(k),u(k)=
∞
k=N
˜
ℓ(x(k),u(k))
ℓT(x(N))
+
N−1
k=0
ℓ(x(k),u(k))
whe e only ˜
ℓappea s in ℓTbecause x(N+k)∈T o
all k≥0 i in Tis posi i e in a ian . Also no e ha
ℓT(x(N)) = ∞
k=N˜
ℓ(x(k),u(k)) can be exp essed in e ms
o only x(N), because x+= (x, κ∞(x)) and u(x)=κ(x)
uniquely de e mine he sequences u(N),u(N+ 1),... and
x(N+ 1),x(N+ 2),... o a gi en x(N). The equi ed
dual-mode con olle esul s i we de e mine a con ol law
u=κ(x) ha esul s in he desi ed p ope ies on some se
Tand hen ea x+= (x, κ(x)+u) by se ing u= 0 on
T.
The ollowing example illus a es Assump ion 1. The ex-
ample is used o all illus a ions in he pape .
Example 1. Conside he sys em (1) wi h
(x, u)=Ax +bu +1
40
x2
1,A=1
2−11
−1−1,b=0
1
cons ain se s
X=x∈R2|−1≤xi≤1,i=1,2
U={u∈R|−1≤u≤1}
s age cos
ℓ(x, u)=1
2x⊤Qx +1
2Ru2,Q=10
01
,R=1/10
e minal cos ℓT(x) = 0, and e minal se Tand con olle
κT:T→R
T=x∈R2|∥x∥2
2≤1,κ
T(x)=0.
I is easy o show ha Tis posi i e in a ian o x(k+
1) = (x(k),κ
T(x(k)). The p oo is gi en in Appendix A
o comple eness.
M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187 183
op imal con ol p oblems (o o he op imiza ion p oblems)
need o be sol ed o ob ain successo ac i e se s. Successo
ac i e se s, which de ine he successo op imal solu ion,
can be cons uc ed bo h o open-loop op imal and closed-
loop op imal solu ions, whe e “closed-loop op imal” e e s
o he usual MPC use o he OCP solu ion on a eceding
ho izon. We no e ha dynamic ela ions o ac i e se s
ha e been used o de e mine obus MPC solu ions o
he linea -quad a ic case (M¨onnigmann and Pannocchia,
2020) and a sh inking ho izon nonlinea case (Dy ska and
M¨onnigmann, 2024) be o e.
Sec ion 2 in oduces he p oblem class. Sec ion 3 s a es
he main esul s. I i s in oduces he symbolic dynamics
in o mally (see (I), (II), (III) in subsec ion 3.3), illus a es
hese ideas wi h a sample p oblem, and s a es o mal
esul s. B ie conclusions and an ou look a e s a ed in
Sec ion 4.
2. PROBLEM STATEMENT
We conside disc e e- ime, nonlinea sys ems o he o m
x(k+ 1) = (x(k),u(k)),k=0,1,... (1a)
wi h s a e x(k)∈Rn, inpu u(k)∈Rm, and a nonlinea
s a e-upda e unc ion :Rn×Rm→Rn. We assume
s a es and inpu s a e subjec o cons ain s
x(k)∈X⊂Rn,
u(k)∈U⊂Rm(1b)
o all ime s eps k, whe e Xand Ua e compac se s
ha con ain he o igin in hei in e io and ha can be
desc ibed as he in e sec ion o a ini e numbe o suble el
se s. Fu he mo e, we assume is wice con inuously
di e en iable on an open supe se o X×U, and (0,0) =
0.
We a e in e es ed in he in ini e-ho izon nonlinea op imal
con ol p oblem o (1)
min
u(k),x(k+1),k=0,1,...
∞
k=0
ℓ(x(k),u(k)) (2a)
subjec o
x(k+ 1) = (x(k),u(k)),k=0,1,...
x(k)∈X,k=0,1,...
u(k)∈U,k=0,1,...
(2b)
o gi en cu en s a e x(0), and he s age cos ℓ(x, u),
which is speci ied la e . The ini e-ho izon p oblem o e
Ns eps
min
u(k),x(k+1),k=0,...N−1ℓT(x(N)) +
N−1
k=0
ℓ(x(k),u(k)) (3a)
subjec o
x(k+ 1) = (x(k),u(k)),k=0, ..., N −1
x(k)∈X,k=0, ..., N −1
u(k)∈U,k=0, ..., N −1
x(N)∈T.
(3b)
is used as an auxilia y p oblem, whe e ℓT(x) and T ⊆X
wi h 0 ∈in Ta e a e minal cos and cons ain , espec-
i ely, which a e desc ibed in mo e de ail in Assump ion 1.
Le FN e e o he se o ini ial s a es x(0) o which (3)
wi h ho izon Nhas a solu ion.
The ollowing Assump ion 1 allows o ela e solu ions
o (2) and (3). The assump ion is mild o linea sys ems
bu s ong o nonlinea sys ems. I he e o e dese es
some commen s, which a e s a ed below.
Assump ion 1. Assume he e exis a se T ⊆X,a
eedback law κ:T →Uand a e minal cos ℓT:T→R
such ha in Tis posi i e in a ian o he closed-loop
sys em and such ha , o any x(0) ∈T, e alua ing
u(k)=κ(x(k)),
x(k+ 1) = (x(k),κ(x(k))) (4)
o k=0,1,... yields a minimum o (2) o ini ial
condi ion x(0) wi h cos ∞
k=0 ℓ(x(k),u(k)) = ℓT(x(0)).
Assump ion 1 is s ong because i essen ially s a es we
know an op imal solu ion o he in ini e-ho izon p oblem
o all ini ial condi ions om Tand know he co e-
sponding in ini e-ho izon cos . This assump ion can eas-
ily be ul illed o linea -quad a ic p oblems unde mild
addi ional condi ions (Chmielewski and Manousiou hakis,
1996; Scokae and Rawlings, 1998), (Bempo ad e al.,
2002, Sec . 3). A nonlinea class ha espec s Assump-
ion 1 esul s o swi ching cos unc ions o he o m
ℓ(x, u)=˜
ℓ(x, u) i x∈T
ˆ
ℓ(x, u) o he wise.
This swi ching cos unc ion class yields he ollowing
ela ion o he in ini e-ho izon cos (2a) o he ini e-
ho izon cos (3a):
∞
k=0
ℓ(x(k),u(k)=
∞
k=N
˜
ℓ(x(k),u(k))
ℓT(x(N))
+
N−1
k=0
ℓ(x(k),u(k))
whe e only ˜
ℓappea s in ℓTbecause x(N+k)∈T o
all k≥0 i in Tis posi i e in a ian . Also no e ha
ℓT(x(N)) = ∞
k=N˜
ℓ(x(k),u(k)) can be exp essed in e ms
o only x(N), because x+= (x, κ∞(x)) and u(x)=κ(x)
uniquely de e mine he sequences u(N),u(N+ 1),... and
x(N+ 1),x(N+ 2),... o a gi en x(N). The equi ed
dual-mode con olle esul s i we de e mine a con ol law
u=κ(x) ha esul s in he desi ed p ope ies on some se
Tand hen ea x+= (x, κ(x)+u) by se ing u= 0 on
T.
The ollowing example illus a es Assump ion 1. The ex-
ample is used o all illus a ions in he pape .
Example 1. Conside he sys em (1) wi h
(x, u)=Ax +bu +1
40
x2
1,A=1
2−11
−1−1,b=0
1
cons ain se s
X=x∈R2|−1≤xi≤1,i=1,2
U={u∈R|−1≤u≤1}
s age cos
ℓ(x, u)=1
2x⊤Qx +1
2Ru2,Q=10
01
,R=1/10
e minal cos ℓT(x) = 0, and e minal se Tand con olle
κT:T→R
T=x∈R2|∥x∥2
2≤1,κ
T(x)=0.
I is easy o show ha Tis posi i e in a ian o x(k+
1) = (x(k),κ
T(x(k)). The p oo is gi en in Appendix A
o comple eness.
3. SYMBOLIC DYNAMICS FOR ACTIVE SETS
3.1 P elimina ies
We need o s a e a clea ela ion be ween solu ions o he
p oblems wi h ini e and in ini e ho izons.
Lemma 1. Conside a cons ained sys em (1) and sup-
pose Assump ion 1 holds.
(a) Le x(0) be an a bi a y ini ial condi ion such ha he
ho izon-Np oblem (3) has an op imal solu ion, which we
deno e
(u(k),x(k+ 1))N−1
k=0 .(5)
I x(N)∈in T, hen (5) ex ended by (4) o all k≥N
is an op imal solu ion o he in ini e-ho izon p oblem (2)
wi h ini ial condi ion x(0).
(b) Con e sely, le x(0) be an a bi a y ini ial condi ion
such ha he in ini e-ho izon p oblem (2) has an op imal
solu ion, which we deno e
(u(k),x(k+ 1))∞
k=0 .(6)
I he e exis s an Nsuch ha x(N)∈in T, hen (6)
unca ed a e Ns eps is an op imal solu ion o (3) wi h
ho izon Nand ini ial condi ion x(0).
The p oo o Lemma 1 is s a ed in Appendix B. No e ha
Lemma 1 canno be s a ed as an equi alence. P oblems (2)
and (3) a e no equi alen because he se o admissible
ini ial condi ions is in gene al la ge o (2) han o (3).
Rema k 1. The condi ion x(N)∈in Tcan nei he be
omi ed no eplaced by x(N)∈T in pa (a) o Lemma 1.
An example whe e all condi ions o Lemma 1 bu x(N)∈
in Thold and he implica ion o Lemma 1 does no hold
is gi en in (M¨onnigmann, 2019, Example 2 and Fig. 3) o
a linea example o which he lemma applies. We s ess
his coun e example is no a pa hological o o he wise
a i icial case. We mus expec ull-dimensional egions o
ini ial condi ions o exis such ha (3) has solu ions wi h
x(N)∈in T ha canno be ex ended o solu ions o he
in ini e-ho izon p oblem (2) e en i Assump ion 1 holds.
3.2 O de ing he cons ain s
We assume wi hou es ic ion he cons ain s a e o de ed
s age by s age, i.e., in he o de
x(0) ∈X,u(0) ∈U,(qs age cons ain s)
x(1) ∈X,u(1) ∈U,(qs age cons ain s)
.
.
.
x(N−1) ∈X,u(N−1) ∈U,(qs age cons ain s)
x(N)∈T (qTcons ain s)
(7)
whe e he numbe o cons ain s pe line is gi en in
pa en heses. We e e o he s ages e iden om (7) as
s age 0, s age 1,..., s age N−1, and he e minal s age,
espec i ely.
I u ns ou o be con enien o deno e se s o ac i e
cons ain s wi h bi sequences. Fo example, conside a
p oblem wi h ho izon N= 2, qs age = 5, qT= 3,
q=Nqs age +qT= 13 cons ain s in he o de (7) and
assume he ac i e se A={1,3,12}appea s. Le qdeno e
he o al numbe o cons ain s q=Nq
s age +qT.
We iden i y Awi h
10100
α0
.00000
α1
.010
α2
(8)
whe e do s a e in oduced o sepa a e s ages and he αi
e e o he bi upels o he s ages. We call (8) an ac i e
se o a oid ph ases like ’ he bi sequence iden i ied wi h
he ac i e se ’. Recall Ais an ac i e se o (3) i he e exis s
an x(0) ∈F
Nwi h ac i e se Aa he op imal solu ion.
3.3 In o mal summa y
We in end o cha ac e ize he solu ion o he ho izon-N+1
p oblem (3) by ca ying o e as much in o ma ion om he
ho izon-Np oblem as possible. We s ess ha he solu ion
o ho izon Nis in gene al no con ained in he solu ion o
ho izon N+ 1. This can easily be illus a ed wi h explici
solu ions in he linea -quad a ic case, whe e he explici
con ol law o ho izon N+ 1 does no coincide wi h he
law o N(whe e e hey bo h exis , i.e., on he easible
se o N; see M¨onnigmann (2019), Fig. 1 o an example).
In ac , he solu ion o an ini ial condi ion x(0) and
ho izon Ndoes emain he same o N+ 1 i x(N)∈
in T. Solu ions o which x(N)∈Tis ul illed wi h
x(N)∈∂T, whe e ∂T e e s o he bounda y o T, do
no ha e his p ope y. I may appea pedan ic o e en
pay a en ion o solu ions wi h x(N)∈∂T, since ∂Tis
a se wi h measu e ze o. Solu ions wi h his p ope y do
in gene al exis , howe e , o ull-dimensional egions o
ini ial condi ions (see M¨onnigmann, 2019, Fig. 3, o an
example again) and he e o e a e no negligible, because
hey cons i u e pa s o FNwi h nonze o measu e.
Ra he han analyzing he e ec o an inc easing ho izon
poin -by-poin , i.e., x(0) by x(0), i p o ed use ul o ca y
ou his analysis wi h he ac i e se s and he egional
op imal eedback laws hey de ine. Essen ially, we need
h ee simple ope a ions on ac i e se s, which a e e e ed
o as I, II and III below. Le ”x” e e o an unspeci ied
bi in an ac i e se . Colo is used as guide o he eye. In
(I) and (III) (bu no in (II)) he e minal cons ain s a e
assumed o be inac i e as e iden om he ailing ze os:
(I) Ex ending an Awi h an inac i e s age
An ac i e se o (3) wi h ho izon Nmay be ex ended
o he igh wi h a ze o s age o ob ain an ac i e se
o (3) wi h ho izon N+ 1:
AN=α0.α1.···.αN−1.0...0
↓
AN+1 =α0.α1.....α
N−1.0...0.0...0
I ANis an ac i e se o ho izon N, hen AN+1 is an
ac i e se o ho izon N+1. Speci ically, AN+1 de ines
he same egion as ANo a supe se he eo , and he
op imal solu ion is equal on he common egion. (See
P op. 1 and Example 2 below o a concise s a emen
o I and an example, espec i ely.)
(II) Dele ing he i s s age o an A
Dele ing a s age on he le in an ac i e se o (3)
wi h ho izon N esul s in an ac i e se o (3) wi h
ho izon N−1:
AN=x...x.x...x.···.x...x.0...0
↓
AN−1=x...x.···.x...x.0...0
184 M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187
I ANis an ac i e se o ho izon N, hen AN−1is
an ac i e se o ho izon N−1. (See P op. 2 and
Example 3.)
While (I) and (II) ela e ac i e se s o di e en ho izons,
combining he wo esul s in a simple ela ion be ween
ac i e se s o he same ho izon.
(III) Symbolic dynamics o ac i e se s
Applying (I) and (II) o an ac i e se o ho izon N
esul s in a new ac i e se o ho izon N ha is pa
o he solu ion:
AN=x...x.x...x.···.x...x.0...0
↓
A′
N=x...x.···.x...x.0...0.0...0
Mo eo e , A′
Nde ines he successo egion o he
egion de ined by AN. In o he wo ds, A′
Nde ines
he egion o which he nominal sys em is d i en when
he i s op imal con ol signal is applied. (See P op. 3
and Example 4.)
3.4 P ecise s a emen s and illus a ions
P oposi ion 1 s a es claim (I) mo e p ecisely. The p opo-
si ion 1 ex ends Lemma 3 om M¨onnigmann (2019) o
linea -quad a ic p oblems o he nonlinea sys em class
ea ed he e.
P oposi ion 1. (a) Conside (3) wi h cons ain o -
de (7). I
α0.··· .αN−1.0···0
qT
(9)
is an ac i e se o (3) wi h ho izon N, hen
α0.··· .αN−1.0···0
l(qX+qU)
.0···0
qT
(10)
is an ac i e se o ho izon N+l o all l≥0.
(b) The se de ined by (10) is equal o o a supe se o he
se de ined by (9). Bo h ac i e se s yield he same solu ion
on he se de ined by (9).
P oo . (a) Acco ding o Lemma 1 pa (a), he ac i e
se (9) can be ex ended wi h (4) o a solu ion o he
in ini e-ho izon p oblem. Since in Tis posi i e in a ian
o he sys em subjec o u(k)=κ(x(k)) om (4) by
assump ion, all cons ain s on u(k), x(k), k≥Na e
inac i e. Acco ding o pa (b) o Lemma 1, he in ini e-
ho izon solu ion can be unca ed o any leng h N+l,
l≥0. Since he cons ain s a e inac i e o all s ages l≥0,
he ac i e se (10) exis s o all l≥0. (b) Le x(0) ∈Rnbe
an a bi a y poin such ha he e exis s a minimum o (3)
wi h ho izon Nand wi h ac i e se AN. Le (5) e e o
his solu ion. Acco ding o P op. 1 his solu ion can be
ex ended o a solu ion (6) o he in ini e-ho izon p oblem.
The posi i e in a iance o in Timplies x(N+l)∈in T
o all l≥0. The e o e (6) can be unca ed o any l≥0
acco ding o pa (b) o Lemma 1. Since his unca ed
solu ion yields he same ac i e cons ain s o he i s N
s ages as he ho izon-Np oblem and solu ion, and since
all addi ional s ages N+l,l≥0 a e inac i e, he ac i e
se (10) esul s o ho izon N+l o any l≥0. We showed
ha any x(0) ∈Rn ha is a solu ion o (3) wi h ho izon
Nand ac i e se (9) also is a solu ion o (3) wi h ho izon
N+land ac i e se (10) o any l≥0.
We illus a e claim (I) and P op. 1 wi h Example 2. The
esul s shown in Figs. 1 o 4 a e ob ained by sol ing (3)
on a g id o 100 ×100 poin s and eco ding all solu ions
including he ac i e se s.
Example 2. Figu e 1 shows all ini ial s a es x(0) ha
esul in he ac i e se s
000001.000000.0(N= 2) (11a)
000001.000000.000000.0(N= 3) (11b)
o (3) wi h Example 1 and N= 2 and N= 3, espec i ely.
I is e iden om he igu es ha he se de ined by (11b)
and N= 3 is a supe se o he se de ined by (11a).
Mo eo e , i is e iden he op imal signal u(0) = −1 esul s
in bo h cases. The emaining op imal inpu signals a e also
equal, which we claim wi hou showing hem. The ac i e
cons ain in bo h ac i e se s co esponds o u(0) = −1.
As a side-e ec , Example 2 shows ha se s o poin s
de ined by an ac i e se may no be connec ed in he
nonlinea case.
P oposi ion 2 and Example 3 belong o claim (II). No e
ha he e minal cons ain s a e no equi ed o be in-
ac i e in P op. 2. Fu he mo e, no e ha claim (II) and
P oposi ion 2 essen ially s a e he p inciple o op imali y
o ac i e se s.
P oposi ion 2. Conside (3) wi h cons ain o de (7). I
AN=α0.α1··· .αN−1.αN(12)
is an ac i e se o (3) wi h ho izon N, hen
AN−1=α1··· .αN−1.αN(13)
is an ac i e se o (3) wi h ho izon N−1. Mo eo e ,
any x(0) om he se de ined by ANis mapped o a
successo s a e in he se de ined by AN−1 o he closed-
loop nominal sys em.
P oo . Le x(0) be any ini ial condi ion such ha (3)
o ho izon N esul s in an op imal solu ion wi h ac i e
se (12). Le he op imal solu ion be deno ed as in (5).
Then he sequence ha esul s om (5) a e emo ing
u(0) and x(1), i.e.,
(u(k),x(k+ 1))N−1
k=1 (14)
is an op imal solu ion o (3) wi h ho izon N−1 and ini ial
condi ion x(1) by he p inciple o op imali y. Since he
same cons ain s a e ac i e o (14) as o (5) he ac i e
se (13) esul s, which p o es he i s pa o he claim.
Since x(0) was an a bi a y ini ial condi ion om he se
de ined by (12), and since x(1) belongs o he se de ined
by (13), he second pa o he claim holds.
Example 3. Figu e 2 shows all ini ial s a es x(0) o
which (3) wi h Example 1 esul s in he ac i e se s
000001.000001.000000.0(N= 3)
000001.000000.0(N= 2)
o N= 2 and N= 3, espec i ely. The yellow egion
in he igu e o N= 2 esul s om he yellow egion o
N= 3 wi h he p inciple o op imali y.
Finally, P op. 3 and Example 4 s a e espec i ely illus a e
he main esul , i.e., he symbolic dynamics o ac i e se s
s a ed in claim (III). I is he e y poin o he pape
o show ha sequences o ac i e se s (e.g., (15)→(16) in
P op. 3 o (18a)→(18b)→(18c) and (19a)→(19b)→(19c)
in Example 4) can be in e ed om an ac i e se wi h
M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187 185
I ANis an ac i e se o ho izon N, hen AN−1is
an ac i e se o ho izon N−1. (See P op. 2 and
Example 3.)
While (I) and (II) ela e ac i e se s o di e en ho izons,
combining he wo esul s in a simple ela ion be ween
ac i e se s o he same ho izon.
(III) Symbolic dynamics o ac i e se s
Applying (I) and (II) o an ac i e se o ho izon N
esul s in a new ac i e se o ho izon N ha is pa
o he solu ion:
AN=x...x.x...x.···.x...x.0...0
↓
A′
N=x...x.···.x...x.0...0.0...0
Mo eo e , A′
Nde ines he successo egion o he
egion de ined by AN. In o he wo ds, A′
Nde ines
he egion o which he nominal sys em is d i en when
he i s op imal con ol signal is applied. (See P op. 3
and Example 4.)
3.4 P ecise s a emen s and illus a ions
P oposi ion 1 s a es claim (I) mo e p ecisely. The p opo-
si ion 1 ex ends Lemma 3 om M¨onnigmann (2019) o
linea -quad a ic p oblems o he nonlinea sys em class
ea ed he e.
P oposi ion 1. (a) Conside (3) wi h cons ain o -
de (7). I
α0.··· .αN−1.0···0
qT
(9)
is an ac i e se o (3) wi h ho izon N, hen
α0.··· .αN−1.0···0
l(qX+qU)
.0···0
qT
(10)
is an ac i e se o ho izon N+l o all l≥0.
(b) The se de ined by (10) is equal o o a supe se o he
se de ined by (9). Bo h ac i e se s yield he same solu ion
on he se de ined by (9).
P oo . (a) Acco ding o Lemma 1 pa (a), he ac i e
se (9) can be ex ended wi h (4) o a solu ion o he
in ini e-ho izon p oblem. Since in Tis posi i e in a ian
o he sys em subjec o u(k)=κ(x(k)) om (4) by
assump ion, all cons ain s on u(k), x(k), k≥Na e
inac i e. Acco ding o pa (b) o Lemma 1, he in ini e-
ho izon solu ion can be unca ed o any leng h N+l,
l≥0. Since he cons ain s a e inac i e o all s ages l≥0,
he ac i e se (10) exis s o all l≥0. (b) Le x(0) ∈Rnbe
an a bi a y poin such ha he e exis s a minimum o (3)
wi h ho izon Nand wi h ac i e se AN. Le (5) e e o
his solu ion. Acco ding o P op. 1 his solu ion can be
ex ended o a solu ion (6) o he in ini e-ho izon p oblem.
The posi i e in a iance o in Timplies x(N+l)∈in T
o all l≥0. The e o e (6) can be unca ed o any l≥0
acco ding o pa (b) o Lemma 1. Since his unca ed
solu ion yields he same ac i e cons ain s o he i s N
s ages as he ho izon-Np oblem and solu ion, and since
all addi ional s ages N+l,l≥0 a e inac i e, he ac i e
se (10) esul s o ho izon N+l o any l≥0. We showed
ha any x(0) ∈Rn ha is a solu ion o (3) wi h ho izon
Nand ac i e se (9) also is a solu ion o (3) wi h ho izon
N+land ac i e se (10) o any l≥0.
We illus a e claim (I) and P op. 1 wi h Example 2. The
esul s shown in Figs. 1 o 4 a e ob ained by sol ing (3)
on a g id o 100 ×100 poin s and eco ding all solu ions
including he ac i e se s.
Example 2. Figu e 1 shows all ini ial s a es x(0) ha
esul in he ac i e se s
000001.000000.0(N= 2) (11a)
000001.000000.000000.0(N= 3) (11b)
o (3) wi h Example 1 and N= 2 and N= 3, espec i ely.
I is e iden om he igu es ha he se de ined by (11b)
and N= 3 is a supe se o he se de ined by (11a).
Mo eo e , i is e iden he op imal signal u(0) = −1 esul s
in bo h cases. The emaining op imal inpu signals a e also
equal, which we claim wi hou showing hem. The ac i e
cons ain in bo h ac i e se s co esponds o u(0) = −1.
As a side-e ec , Example 2 shows ha se s o poin s
de ined by an ac i e se may no be connec ed in he
nonlinea case.
P oposi ion 2 and Example 3 belong o claim (II). No e
ha he e minal cons ain s a e no equi ed o be in-
ac i e in P op. 2. Fu he mo e, no e ha claim (II) and
P oposi ion 2 essen ially s a e he p inciple o op imali y
o ac i e se s.
P oposi ion 2. Conside (3) wi h cons ain o de (7). I
AN=α0.α1··· .αN−1.αN(12)
is an ac i e se o (3) wi h ho izon N, hen
AN−1=α1··· .αN−1.αN(13)
is an ac i e se o (3) wi h ho izon N−1. Mo eo e ,
any x(0) om he se de ined by ANis mapped o a
successo s a e in he se de ined by AN−1 o he closed-
loop nominal sys em.
P oo . Le x(0) be any ini ial condi ion such ha (3)
o ho izon N esul s in an op imal solu ion wi h ac i e
se (12). Le he op imal solu ion be deno ed as in (5).
Then he sequence ha esul s om (5) a e emo ing
u(0) and x(1), i.e.,
(u(k),x(k+ 1))N−1
k=1 (14)
is an op imal solu ion o (3) wi h ho izon N−1 and ini ial
condi ion x(1) by he p inciple o op imali y. Since he
same cons ain s a e ac i e o (14) as o (5) he ac i e
se (13) esul s, which p o es he i s pa o he claim.
Since x(0) was an a bi a y ini ial condi ion om he se
de ined by (12), and since x(1) belongs o he se de ined
by (13), he second pa o he claim holds.
Example 3. Figu e 2 shows all ini ial s a es x(0) o
which (3) wi h Example 1 esul s in he ac i e se s
000001.000001.000000.0(N= 3)
000001.000000.0(N= 2)
o N= 2 and N= 3, espec i ely. The yellow egion
in he igu e o N= 2 esul s om he yellow egion o
N= 3 wi h he p inciple o op imali y.
Finally, P op. 3 and Example 4 s a e espec i ely illus a e
he main esul , i.e., he symbolic dynamics o ac i e se s
s a ed in claim (III). I is he e y poin o he pape
o show ha sequences o ac i e se s (e.g., (15)→(16) in
P op. 3 o (18a)→(18b)→(18c) and (19a)→(19b)→(19c)
in Example 4) can be in e ed om an ac i e se wi h
Fig. 1. Illus a ion o claim (I) and P op. 1. The egion
de ined by (11a) o N= 2 (yellow, op diag ams) is a
subse o he egion de ined by (11b) o N= 3 (g een
and yellow, bo om diag ams, yellow used o highligh
he egion om N= 2). G ey a ea delinea es FN.
inac i e e minal cons ain s, whe e hese sequences o
ac i e se s de ine sequences o egions h ough which he
sys ems e ol es unde MPC. Ob aining hese does no
equi e sol ing op imal con ol p oblems, bu he successo
ac i e se s can simply be cons uc ed by dele ing he i s
Fig. 2. Illus a ion o claim (II) and P op. 2. Ini ial s a es
om he egion de ined by (12) o N= 3 p opaga e
o he egion de ined by (13) o N= 2.
s age o he gi en ac i e se and appending i by an
inac i e s age.
P oposi ion 3. Conside (3) wi h cons ain o de (7).
I (15) is an ac i e se , hen (16) is an ac i e se , whe e (16)
esul s om dele ing α0and inse ing an inac i e penul i-
ma e s age in (15).
186 M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187
Fig. 3. Illus a ion o claim (III) and P op. 3. The egions
belong o he ac i e se s (18). Any ini ial poin x(0)
loca ed in he ed egion passes h ough he cyan and
g een egion and subsequen ly en e s he blue egion.
The sequence o hese ac i e se s can be ound om
he i s ac i e se (18a) wi h claim (III) and P op. 3
wi hou sol ing any op imal con ol p oblem.
α0.α1.α2.··· .αN−1.0···0
qT
(15)
α1.α2.··· .αN−1.0···0.0···0
qT
(16)
Mo eo e , (16) de ines he successo se o he se de ined
by (15) o a supe se he eo . In o he wo ds, o any x(0)
ha belongs o he se de ined by (15), he successo s a e
belongs o he se de ined by (16).
P oo . I (15) is an ac i e se o ho izon N, hen
α1.α2.··· .αN−1.0···0 (17)
is an ac i e se o ho izon Nacco ding o P op. 2.
Inse ing an inac i e penul ima e inac i e s age in (17)
esul s in (16), which is an ac i e se o ho izon N
acco ding o P op. 1.
Example 4. Conside he op imal con ol p oblem (3) o
Example 1 and N= 3.
Figu e 3 shows he egions de ined by ac i e se s
000001.000010.000000.0 ( ed) (18a)
000010.000000.000000.0 (cyan) (18b)
000000.000000.000000.0 (g een) (18c)
which esul om applying claim (III) o P op. 3 o (18a)
once and wice. The colo s s a ed in pa en heses co e-
spond o he colo s o he egions in Fig. 3. To show
ano he example, Fig. 4 shows he egions de ined by ac i e
se s
Fig. 4. Illus a ion o claim (III) and P op. 3 wi h ano he
example. The egions belong o he ac i e se s (19).
See he cap ion o Fig. 3 o u he explana ions.
000001.000001.000000.0 ( ed) (19a)
000001.000000.000000.0 (cyan) (19b)
000000.000000.000000.0 (g een) (19c)
which esul om applying claim (III) o P op. 3 o (19a)
once and wice.
4. CONCLUSIONS AND OUTLOOK
We showed how o in e ac i e se s ha cha ac e ize he
op imal successo solu ion gi en he ac i e se o he
cu en op imal solu ion. I was he pu pose o he pape
o es ablish he ope a ions ha need o be ca ied ou
wi h he ac i e se s. Because hese ules do no in ol e
op imal con ol (o o he op imiza ion) p oblems bu a e
based on simple ope a ions like index shi s, we coined he
e m symbolic dynamics o ac i e se s o hem.
Fu u e wo k will ocus on ex ending he p oblem class. Ex-
ensions may ei he a emp o d op some o he assump-
ions, o o combine he wo-s age cos unc ion wi h an
ad anced e minal con olle such as an economic model
p edic i e con olle .
Appendix A. PROPERTIES OF EXAMPLE 1
The linea au onomous pa o he dynamical sys em is
locally asymp o ically s able a x= 0, since i s eigen alues
a e λ1,2=−1
2±i1
2. Fu he mo e, i is easy o show
ha ∥ξ(k+ 1)∥2=∥Aξ(k)∥2=1
2∥ξ(k)∥2 o he linea
au onomous pa o he example. This implies ∥x(k+
1)∥2=∥Ax(k)+1
40
x2
1∥2≤1
2∥x(k)∥2+1
42x4
1. Since he
M. Mönnigmann e al. / IFAC Pape sOnLine 58-18 (2024) 181–187 187
Fig. 3. Illus a ion o claim (III) and P op. 3. The egions
belong o he ac i e se s (18). Any ini ial poin x(0)
loca ed in he ed egion passes h ough he cyan and
g een egion and subsequen ly en e s he blue egion.
The sequence o hese ac i e se s can be ound om
he i s ac i e se (18a) wi h claim (III) and P op. 3
wi hou sol ing any op imal con ol p oblem.
α0.α1.α2.··· .αN−1.0···0
qT
(15)
α1.α2.··· .αN−1.0···0.0···0
qT
(16)
Mo eo e , (16) de ines he successo se o he se de ined
by (15) o a supe se he eo . In o he wo ds, o any x(0)
ha belongs o he se de ined by (15), he successo s a e
belongs o he se de ined by (16).
P oo . I (15) is an ac i e se o ho izon N, hen
α1.α2.··· .αN−1.0···0 (17)
is an ac i e se o ho izon Nacco ding o P op. 2.
Inse ing an inac i e penul ima e inac i e s age in (17)
esul s in (16), which is an ac i e se o ho izon N
acco ding o P op. 1.
Example 4. Conside he op imal con ol p oblem (3) o
Example 1 and N= 3.
Figu e 3 shows he egions de ined by ac i e se s
000001.000010.000000.0 ( ed) (18a)
000010.000000.000000.0 (cyan) (18b)
000000.000000.000000.0 (g een) (18c)
which esul om applying claim (III) o P op. 3 o (18a)
once and wice. The colo s s a ed in pa en heses co e-
spond o he colo s o he egions in Fig. 3. To show
ano he example, Fig. 4 shows he egions de ined by ac i e
se s
Fig. 4. Illus a ion o claim (III) and P op. 3 wi h ano he
example. The egions belong o he ac i e se s (19).
See he cap ion o Fig. 3 o u he explana ions.
000001.000001.000000.0 ( ed) (19a)
000001.000000.000000.0 (cyan) (19b)
000000.000000.000000.0 (g een) (19c)
which esul om applying claim (III) o P op. 3 o (19a)
once and wice.
4. CONCLUSIONS AND OUTLOOK
We showed how o in e ac i e se s ha cha ac e ize he
op imal successo solu ion gi en he ac i e se o he
cu en op imal solu ion. I was he pu pose o he pape
o es ablish he ope a ions ha need o be ca ied ou
wi h he ac i e se s. Because hese ules do no in ol e
op imal con ol (o o he op imiza ion) p oblems bu a e
based on simple ope a ions like index shi s, we coined he
e m symbolic dynamics o ac i e se s o hem.
Fu u e wo k will ocus on ex ending he p oblem class. Ex-
ensions may ei he a emp o d op some o he assump-
ions, o o combine he wo-s age cos unc ion wi h an
ad anced e minal con olle such as an economic model
p edic i e con olle .
Appendix A. PROPERTIES OF EXAMPLE 1
The linea au onomous pa o he dynamical sys em is
locally asymp o ically s able a x= 0, since i s eigen alues
a e λ1,2=−1
2±i1
2. Fu he mo e, i is easy o show
ha ∥ξ(k+ 1)∥2=∥Aξ(k)∥2=1
2∥ξ(k)∥2 o he linea
au onomous pa o he example. This implies ∥x(k+
1)∥2=∥Ax(k)+1
40
x2
1∥2≤1
2∥x(k)∥2+1
42x4
1. Since he
las exp ession is no la ge han 1
2+1
16 o all x(k) he
closed uni disk, he example is posi i e in a ian on T o
u=κT(x). I can be shown o be posi i e in a ian on he
in e io o Tby he same a gumen s.
Appendix B. PROOF OF LEMMA 1
Pa (a): Conside he in ini e-ho izon p oblem (2) o
x(0), assume (5) ex ended by (4) is no an op imal solu ion
and show a con adic ion esul s. I (5) ex ended by
(4) is no an op imal solu ion, hen he e exis s, in any
neighbo hood o (5) ex ended by (4), a di e en sequence
ha espec s he cons ain s o (2) and yields a lowe cos
unc ion alue o (2). Le
(˜u(k),˜x(k+ 1))∞
k=0
e e o his sequence and assume i is om a neighbo hood
o (4, 5) su icien ly small o x(N)∈in T o imply
˜x(N)∈in T.(B.1)
This implies a leas one o he e ms
N−1
k=0
ℓ(˜x(k),˜u(k)) +
∞
k=N
ℓ(˜x(k),˜u(k))
e alua es o a lowe alue han i s coun e pa in
N−1
k=0
ℓ(x(k),u(k)) +
∞
k=N
ℓ(x(k),u(k)).
I his is he case o he second e m, his con adic s he
op imali y o (4). I his is he case o he i s e m, his
con adic s he op imali y o (5), whe e we used he ac
ha
(˜u(k),˜x(k+ 1))N−1
k=0
is admissible o (3) because i espec s (B.1) and he
emaining cons ain s o (3) a e ul illed because (3) has
hem in common wi h (2). Thus, he desi ed con adic ion
esul s in any case.
Pa (b): The unca ed sequence (6) is easible o he
ho izon-Np oblem (3), since p oblems (2) and (3) ha e
he cons ain s o s ages k=0,...,N −1 in common,
and he only emaining cons ain o (3), i.e., x(N)∈T,
is ul illed by assump ion. Now assume he unca ed
sequence is easible bu no op imal o (3). This implies
he e exis s (in any, a bi a ily small, neighbo hood N⊂
RmN ×RnN o he unca ed sequence) a di e en sequence
(¯u(k),¯x(k+ 1))N−1
k=0 (B.2)
ha esul s in a lowe cos unc ion alue, wi h ¯x(N)∈T
(and, in gene al, ¯x(N)=x(N)). By pa (a) we can
ex end (B.2) o a solu ion o he in ini e-ho izon p oblem,
which we deno e (¯u(k),¯x(k))∞
k=0. Since
(¯u(k),¯x(k))N−1
k=0 =(u(k),x(k))N−1
k=0 ,
we also ha e
(¯u(k),¯x(k))∞
k=0 =(u(k),x(k))∞
k=0 ,(B.3)
and he op imali y o he o me does no con adic he
op imali y o he la e .
Since Ncan be chosen o be a bi a ily small, we can
make he di e ence be ween he l.h.s. and .h.s. in (B.3)
a bi a ily small (e.g., in he ob ious 2-no m). This implies
he e exis s a solu ion (¯u(k),¯x(k+ 1))∞
k=0 in any neighbo -
hood o (u(k),x(k+ 1))∞
k=0 such ha he o me esul s in
a lowe cos unc ion alue han he la e , which is he
desi ed con adic ion.
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