Acta mathematica scientia, Series B >
TIME AND NORM OPTIMAL CONTROLS: A SURVEY OF RECENT RESULTS AND OPEN PROBLEMS
Received date: 2011-08-12
Online published: 2011-11-20
We present in this paper a survey of recent results on the relation between time and norm optimality for linear systems and the infinite dimensional version of Pontryagin’s maximum principle. In particular, we discuss optimality (or nonoptimality) of singular controls satisfying the maximum principle and smoothness of the costate in function of smoothness of the target.
H. O. Fattorini . TIME AND NORM OPTIMAL CONTROLS: A SURVEY OF RECENT RESULTS AND OPEN PROBLEMS[J]. Acta mathematica scientia, Series B, 2011 , 31(6) : 2203 -2218 . DOI: 10.1016/S0252-9602(11)60394-9
[1] Bellman R E, Glicksberg I, Gross O A. On the “bang-bang” control problem. Quart Appl Math, 1956, 14: 11–18
[2] Bellman R E. Dynamic Programming. Princeton: Princeton University Press, 1957
[3] Boltyanski V G, Gamkrelidze R V, Pontryagin L S. On the theory of optimal processes. Dokl Akad Nauk SSSR, 1956, 110: 7–10
[4] Boltyanskii V G. Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM J Control, 1966, 4: 326–361
[5] Cannarsa P, Cˇarjâ O. On the Bellman equation for the minimum time problem in infinite dimensions. SIAM J Contr Opt, 2004, 43: 532–548
[6] Cˇarjâ O. The minimal time function in infinite dimension. SIAM J Contr Opt, 1993, 31: 1103–1114
[7] Cˇarjâ O. On the minimum time function and the minimum energy problem; a nonlinear case. Syst Contr Lett, 2006, 55: 543–548
[8] Cˇarjâ O, Lazu A. On the minimal time null controllability of the heat equation. Discrete Contin Dyn Syst, Proc 7th AIMS Conference on Dynamical Systems, Differential Equations and Applications, 2009: 143–150
[9] Fattorini H O. Time-optimal control of solutions of operational differential equations. SIAM J Contr, 1964, 2: 54–59
[10] Fattorini H O. The maximum principle in infinite dimension. Discrete Contin Dyn Syst, 2000, 6: 557–574
[11] Fattorini H O. Time optimality and the maximum principle in infinite dimension. Optimization, 2001, 50: 361–385
[12] Fattorini H O. A survey of the time optimal problem and the norm optimal problem in infinite dimension. Cubo Mat Educational, 2001, 3: 147–169
[13] Fattorini H O. Existence of singular extremals and singular functionals in reachable spaces. J Evolution Equations, 2001, 1: 325–347
[14] Fattorini H O. Vanishing of the costate in Pontryagin’s maximum principle and singular time optimal controls. J Evolution Equations, 2004, 4: 99–123
[15] Fattorini H O. Sufficiency of the maximum principle for time optimality. Cubo: A Mathematical Journal, 2005, 7: 27–37
[16] Fattorini H O. Smoothness of the costate and the target in the time and norm optimal problems. Optimization, 2006, 55: 19–36
[17] Fattorini H O. Infinite Dimensional Linear Control Systems. North-Holland Mathematical Studies 201. Amsterdam: Elsevier, 2005
[18] Fattorini H O. Linear control systems in sequence spaces//Amann H, et al eds. Funct Anal Evol Equ. The G¨unter Lumer Volume. Springer, 2007: 273–290
[19] Fattorini H O. Regular and strongly regular time and norm optimal controls. Cubo: A Mathematical Journal, 2008, 10: 77–92
[20] Fattorini H O. Time and norm optimality of weakly singular controls//Progress in Nonlinear Differential Equations and Their Applications 60 (Parabolic Problems: Herbert Amann Festschrift). Basel AG: Springer, 2011: 233–249
[21] Fattorini H O. Strongly regular time and norm optimal controls. To appear in Dynamics of Continuous, Discrete and Impulsive Systems.
[22] Frankowska H. Value function in optimal control, parts 1, 2. ICTP Lect Notes, 8. Abdus Salam Int Cent Theoret Phys, Trieste, 2002: 516–653
[23] Lin P, Wang G. Blowup time optimal control for ordinary differential equations. SIAM J Contr Optim, 2011, 49: 73–105
[24] Pontryagin L S, Boltyanski V G, Gamkrelidze R V, Mischenko E F. The Mathematical Theory of Optimal Processes. Moscow: Gostejizdat, 1961; English translation, New York: Wiley, 1962
/
| 〈 |
|
〉 |