最短时间控制问题与最小范数控制问题的等价性
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摘要
本文介绍我们得到的关于时间最优控制问题与范数最优控制问题之间等价性的一些最新发展,以及它的一些应用.
This paper presents some of our new developments about the equivalence between time optimal control problems and norm optimal control problems. Besides, several applications of this equivalence are given.
This paper presents some of our new developments about the equivalence between time optimal control problems and norm optimal control problems. Besides, several applications of this equivalence are given.
引文
1 Wiener N.Cybernetics,or Control and Communication in the Animal and the Machine.Cambridge:MIT Press,1948
2 Tsien H S.Engineering Cybernetics.New York:McGraw-Hill Book Company,1954
3 Pontryagin L S,Boltyanski V G,Gamkrelidge R V,et al.The Mathematical Theory of Optimal Processes.New York:John Wiley&Sons,1962
4 Bellman R.Dynamic Programming.Princeton:Princeton University Press,1957
5 Kalman R E.On the general theory of control systems.In:Automatic and Remote Control.Proceedings of the First International Congress of the International Federation of Automatic Control(Moscow,1960).London:Butterworth,1961,481-492
6 Fattorini H O.Time-optimal control of solutions of operational differenital equations.J SIAM Control Ser A,1964,2:54-59
7 Fattorini H O.Infinite Dimensional Linear Control Systems:The Time Optimal and Norm Optimal Problems.NorthHolland Mathematics Studies,vol.201.Amsterdam:Elsevier B V,2005
8 Gozzi F,Loreti P.Regularity of the minimum time function and minimum energy problems:The linear case.SIAMJ Control Optim,1999,37:1195-1221
9 Carjǎ O.On the minimal time function for distributed control systems in Banach spaces.J Optim Theory Appl,1984,44:397-406
10 Carjǎ O.The minimal time function in infinite dimensions.SIAM J Control Optim,1993,31:1103-1114
11 Gyurkovics E.H?lder condition for the minimum time function of linear systems.In:System Modelling and Optimization(Copenhagen,1983).Lecture Notes in Control and Information Sciences,vol.59.Berlin:Springer,1984,383-392
12 Wang G,Zuazua E.On the equivalence of minimal time and minimal norm controls for internally controlled heat equations.SIAM J Control Optim,2012,50:2938-2958
13 Wang G.L∞-null controllability for the heat equation and its consequences for the time optimal control problem.SIAM J Control Optim,2008,47:1701-1720
14 Kunisch K,Wang L.Time optimal control of the heat equation with pointwise control constraints.ESAIM Control Optim Calc Var,2013,19:460-485
15 Kunisch K,Wang L.Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints.J Math Anal Appl,2012,395:114-130
16 Tucsnak M,Wang G,Wu C.Perturbations of time optimal control problems for a class of abstract parabolic systems.SIAM J Control Optim,2016,54:2965-2991
17 Yu H.Approximation of time optimal controls for heat equations with perturbations in the system potential.SIAM JControl Optim,2014,52:1663-1692
18 Yu H.Equivalence of minimal time and minimal norm control problems for semilinear heat equations.Systems Control Lett,2014,73:17-24
19 Zhang C.The time optimal control with constraints of the rectangular type for linear time-varying ODEs.SIAM JControl Optim,2013,51:1528-1542
20 Zhang Y.Two equivalence theorems of different kinds of optimal control problems for Schr?dinger equations.SIAM JControl Optim,2015,53:926-947
21 Wang G,Zhang Y.Decompositions and bang-bang properties.Math Control Relat Fields,2017,7:73-170
22 Qin S,Wang G.Equivalence between minimal time and minimal norm control problems for the heat equation.SIAMJ Control Optim,2018,56:981-1010
23 Wang G,Xu Y,Zhang Y.Attainable subspaces and the bang-bang property of time optimal controls for heat equations.SIAM J Control Optim,2015,53:592-621
24 Wang G,Wang L,Xu Y,et al.Time Optimal Control of Evolution Equations.Progress in Nonlinear Differential Equations and Their Applications,vol.92.Subseries in Control.Cham:Birkh?user/Springer,2018
25 Carjǎ O,Lazu A I.On the continuity of the state constrained minimal time function.Electron J Qual Theory Differ Equ,2014,2014:1-16
26 Durea M,Pant?iruc M,Strugariu R.Minimal time function with respect to a set of directions:Basic properties and applications.Optim Methods Softw,2016,31:535-561
27 Lazu A I.On the regularity of the state constrained minimal time function.An S?tiint?Univ Al I Cuza Ia?si Mat(NS),2011,57:151-160
28 Lu X,Wang L,Yan Q.Computation of time optimal control problems governed by linear ordinary differential equations.J Sci Comput,2017,73:1-25
29 Fabre C,Puel J P,Zuazua E.Approximate controllability of the semilinear heat equation.Proc Roy Soc Edinburgh Sect A,1995,125:31-61
2 Tsien H S.Engineering Cybernetics.New York:McGraw-Hill Book Company,1954
3 Pontryagin L S,Boltyanski V G,Gamkrelidge R V,et al.The Mathematical Theory of Optimal Processes.New York:John Wiley&Sons,1962
4 Bellman R.Dynamic Programming.Princeton:Princeton University Press,1957
5 Kalman R E.On the general theory of control systems.In:Automatic and Remote Control.Proceedings of the First International Congress of the International Federation of Automatic Control(Moscow,1960).London:Butterworth,1961,481-492
6 Fattorini H O.Time-optimal control of solutions of operational differenital equations.J SIAM Control Ser A,1964,2:54-59
7 Fattorini H O.Infinite Dimensional Linear Control Systems:The Time Optimal and Norm Optimal Problems.NorthHolland Mathematics Studies,vol.201.Amsterdam:Elsevier B V,2005
8 Gozzi F,Loreti P.Regularity of the minimum time function and minimum energy problems:The linear case.SIAMJ Control Optim,1999,37:1195-1221
9 Carjǎ O.On the minimal time function for distributed control systems in Banach spaces.J Optim Theory Appl,1984,44:397-406
10 Carjǎ O.The minimal time function in infinite dimensions.SIAM J Control Optim,1993,31:1103-1114
11 Gyurkovics E.H?lder condition for the minimum time function of linear systems.In:System Modelling and Optimization(Copenhagen,1983).Lecture Notes in Control and Information Sciences,vol.59.Berlin:Springer,1984,383-392
12 Wang G,Zuazua E.On the equivalence of minimal time and minimal norm controls for internally controlled heat equations.SIAM J Control Optim,2012,50:2938-2958
13 Wang G.L∞-null controllability for the heat equation and its consequences for the time optimal control problem.SIAM J Control Optim,2008,47:1701-1720
14 Kunisch K,Wang L.Time optimal control of the heat equation with pointwise control constraints.ESAIM Control Optim Calc Var,2013,19:460-485
15 Kunisch K,Wang L.Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints.J Math Anal Appl,2012,395:114-130
16 Tucsnak M,Wang G,Wu C.Perturbations of time optimal control problems for a class of abstract parabolic systems.SIAM J Control Optim,2016,54:2965-2991
17 Yu H.Approximation of time optimal controls for heat equations with perturbations in the system potential.SIAM JControl Optim,2014,52:1663-1692
18 Yu H.Equivalence of minimal time and minimal norm control problems for semilinear heat equations.Systems Control Lett,2014,73:17-24
19 Zhang C.The time optimal control with constraints of the rectangular type for linear time-varying ODEs.SIAM JControl Optim,2013,51:1528-1542
20 Zhang Y.Two equivalence theorems of different kinds of optimal control problems for Schr?dinger equations.SIAM JControl Optim,2015,53:926-947
21 Wang G,Zhang Y.Decompositions and bang-bang properties.Math Control Relat Fields,2017,7:73-170
22 Qin S,Wang G.Equivalence between minimal time and minimal norm control problems for the heat equation.SIAMJ Control Optim,2018,56:981-1010
23 Wang G,Xu Y,Zhang Y.Attainable subspaces and the bang-bang property of time optimal controls for heat equations.SIAM J Control Optim,2015,53:592-621
24 Wang G,Wang L,Xu Y,et al.Time Optimal Control of Evolution Equations.Progress in Nonlinear Differential Equations and Their Applications,vol.92.Subseries in Control.Cham:Birkh?user/Springer,2018
25 Carjǎ O,Lazu A I.On the continuity of the state constrained minimal time function.Electron J Qual Theory Differ Equ,2014,2014:1-16
26 Durea M,Pant?iruc M,Strugariu R.Minimal time function with respect to a set of directions:Basic properties and applications.Optim Methods Softw,2016,31:535-561
27 Lazu A I.On the regularity of the state constrained minimal time function.An S?tiint?Univ Al I Cuza Ia?si Mat(NS),2011,57:151-160
28 Lu X,Wang L,Yan Q.Computation of time optimal control problems governed by linear ordinary differential equations.J Sci Comput,2017,73:1-25
29 Fabre C,Puel J P,Zuazua E.Approximate controllability of the semilinear heat equation.Proc Roy Soc Edinburgh Sect A,1995,125:31-61