no. 1
Time-optimality by distance-optimality for parabolic control systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 1, pp. 79-103

The equivalence of time-optimal and distance-optimal control problems is shown for a class of parabolic control systems. Based on this equivalence, an approach for the efficient algorithmic solution of time-optimal control problems is investigated. Numerical examples are provided to illustrate that the approach works well in practice.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2019046
Classification : 49K20, 49M15
Keywords: Time-optimal controls, bang-bang controls, distance optimal controls, parabolic control systems 
@article{M2AN_2020__54_1_79_0,
     author = {Bonifacius, Lucas and Kunisch, Karl},
     title = {Time-optimality by distance-optimality for parabolic control systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {79--103},
     year = {2020},
     publisher = {EDP Sciences},
     volume = {54},
     number = {1},
     doi = {10.1051/m2an/2019046},
     mrnumber = {4051842},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2019046/}
}
TY  - JOUR
AU  - Bonifacius, Lucas
AU  - Kunisch, Karl
TI  - Time-optimality by distance-optimality for parabolic control systems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2020
SP  - 79
EP  - 103
VL  - 54
IS  - 1
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2019046/
DO  - 10.1051/m2an/2019046
LA  - en
ID  - M2AN_2020__54_1_79_0
ER  - 
%0 Journal Article
%A Bonifacius, Lucas
%A Kunisch, Karl
%T Time-optimality by distance-optimality for parabolic control systems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2020
%P 79-103
%V 54
%N 1
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2019046/
%R 10.1051/m2an/2019046
%G en
%F M2AN_2020__54_1_79_0
Bonifacius, Lucas; Kunisch, Karl. Time-optimality by distance-optimality for parabolic control systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 1, pp. 79-103. doi: 10.1051/m2an/2019046

H. Amann and J. Escher, Analysis. I. Translated from the 1998 German original by Gary Brookfield. Birkhäuser Verlag, Basel (2005). | MR | Zbl

L. Bonifacius, Numerical analysis of parabolic time-optimal control problems. Ph.D. thesis, Technische Universität München (2018).

L. Bonifacius and K. Pieper, Strong stability of linear parabolic time-optimal control problems. ESAIM: COCV 25 (2019) 35. | MR | Numdam | Zbl

L. Bonifacius, K. Pieper and B. Vexler, Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls. SIAM J. Control Optim. 57 (2019) 1730–1756. | MR | Zbl | DOI

E. Casas, D. Wachsmuth and G. Wachsmuth, Sufficient second-order conditions for bang-bang control problems. SIAM J. Control Optim. 55 (2017) 3066–3090. | MR | DOI

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig . Springer-Verlag, Berlin (1992). | MR | Zbl

J.C. Dunn, Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J. Control Optim. 18 (1980) 473–487. | MR | Zbl | DOI

H.O. Fattorini, Infinite dimensional linear control systems. The time optimal and norm optimal problems. In: Vol. 201 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam (2005). | MR | Zbl

R. Glowinski, J.-L. Lions and J. He, Exact and approximate controllability for distributed parameter systems. A numerical approach. In: Vol. of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2008). | MR

F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problems: the linear case. SIAM J. Control Optim. 37 (1999) 1195–1221. | MR | Zbl | DOI

M. Gugat, A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems. Control of partial differential equations (Jacksonville, FL, 1998). J. Comput. Appl. Math. 114 (2000) 103–119. | MR | Zbl | DOI

Q. Han and F.-H. Lin, Nodal sets of solutions of parabolic equations. II. Comm. Pure Appl. Math. 47 (1994) 1219–1238. | MR | Zbl | DOI

H. Hermes and J.P. Lasalle, Functional analysis and time optimal control. In: Vol. 56 of Mathematics in Science and Engineering. Academic Press, New York-London (1969). | MR | Zbl

M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17 (2006) 159–187. | MR | Zbl | DOI

K. Ito and K. Kunisch, Semismooth Newton methods for time-optimal control for a class of ODEs, SIAM J. Control Optim. 48 (2010) 3997–4013. | MR | Zbl | DOI

C.Y. Kaya and J.L. Noakes, Computational method for time-optimal switching control. J. Optim. Theory Appl. 117 (2003) 69–92. | MR | Zbl | DOI

W. Krabs, Optimal control of processes governed by partial differential equations. I. Heating processes. Z. Oper. Res. Ser. A-B 26 (1982) A21–A48. | MR | Zbl

K. Kunisch and D. Wachsmuth, On time optimal control of the wave equation, its regularization and optimality system. ESAIM: COCV 19 (2013) 317–336. | MR | Zbl | Numdam

K. Kunisch, K. Pieper and A. Rund, Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach. ESAIM: M2AN 50 (2016) 381–414. | MR | Numdam | DOI

X.J. Li and J.M. Yong, Optimal control theory for infinite-dimensional systems. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc, Boston, MA (1995). | MR | Zbl

X. Lu, L. Wang and Q. Yan, Computation of time optimal control problems governed by linear ordinary differential equations. J. Sci. Comput. 73 (2017) 1–25. | MR | DOI

J.W. Macki and A. Strauss, Introduction to optimal control theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin (1982). | MR | Zbl

E.-B. Meier and A.E. Bryson Jr, Efficient algorithm for time-optimal control of a two-link manipulator. J. Guidance Control Dynam. 13 (1990) 859–866. | MR | Zbl | DOI

A. Münch and F. Periago, Numerical approximation of bang-bang controls for the heat equation: an optimal design approach. Syst. Control Lett. 62 (2013) 643–655. | MR | Zbl | DOI

A. Münch and E. Zuazua. Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Prob. 26 (2010) 085018, 39. | MR | Zbl | DOI

J. Nocedal and S.J. Wright, Numerical optimization, 2nd edition. In: Springer Series in Operations Research and Financial Engineering. Springer, New York, NY (2006). | MR | Zbl

E.M. Ouhabaz, Analysis of heat equations on domains. In: Vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2005). | MR | Zbl

A. Pazy, Semigroups of linear operators and applications to partial differential equations. In: Vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York (1983). | MR | Zbl

K. Pieper, Finite element discretization and efficient numerical solution of elliptic and parabolic sparse control problems. Ph.D. thesis, Technische Universität München (2015).

S. Qin and G. Wang, Equivalence between minimal time and minimal norm control problems for the heat equation. SIAM J. Control Optim. 56 (2018) 981–1010. | MR | Zbl | DOI

S.M. Robinson, Normal maps induced by linear transformations. Math. Oper. Res. 17 (1992) 691–714. | MR | Zbl | DOI

F. Trltzsch, Optimal control of partial differential equations. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. In: Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). | MR | Zbl | DOI

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2009). | MR | Zbl

W. Wang and M.Á. Carreira-Perpiñán, Projection onto the probability simplex: an efficient algorithm with a simple proof, and an application. Preprint arXiv:1309.1541 (2013).

G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications. SIAM J. Control Optim. 51 (2013) 848–880. | MR | Zbl | DOI

G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations. SIAM J. Control Optim. 50 (2012) 2938–2958. | MR | Zbl | DOI

G. Wang, L. Wang, Y. Xu and Y. Zhang, Time optimal control of evolution equations. Progress in Nonlinear Differential Equations and Their Applications, Springer International Publishing (2018). | MR | DOI

C. Zhang, The time optimal control with constraints of the rectangular type for linear time-varying ODEs. SIAM J. Control Optim. 51 (2013) 1528–1542. | MR | Zbl | DOI

Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations. SIAM J. Control Optim. 53 (2015) 926–947. | MR | Zbl | DOI

Cité par Sources :