Time optimal control problems for some non-smooth systems. (English) Zbl 1307.49018
Summary: Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin’s maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin’s maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one gets a chance to get Pontryagin’s maximum principle for the original optimal classical control problem. Existence results are also considered.
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
optimal control problems; Pontryagin’s maximum principle; optimal relaxed control; optimal blowup time; optimal quenching timeReferences:
| [1] | C. Bandle, Blowup in diffusion equations: A survey,, J. Comput. Appl. Math., 97, 3 (1998) · Zbl 0932.65098 · doi:10.1016/S0377-0427(98)00100-9 |
| [2] | C. Y. Chan, New results in quenching,, in World Congress of Nonlinear Analysts ’92, 427 (1992) · Zbl 0847.35063 |
| [3] | C. Y. Chan, Quenching for semilinear singular parabolic problems,, SIAM J. Math. Anal., 20, 558 (1989) · Zbl 0678.35056 · doi:10.1137/0520039 |
| [4] | M. Escobedo, Boundedness and blow up for a semilinear reaction diffusion system,, J. Differential Equations, 89, 176 (1991) · Zbl 0735.35013 · doi:10.1016/0022-0396(91)90118-S |
| [5] | R. Glassey, Blow-up theorems for nonlinear wave-equations,, Math. Z., 132, 183 (1973) · Zbl 0247.35083 · doi:10.1007/BF01213863 |
| [6] | J. S. Guo, The profile near quenching time for the solution of a singular semilinear heat equation,, Proc. Edinburgh Math. Soc., 40, 437 (1997) · Zbl 0903.35007 · doi:10.1017/S0013091500023932 |
| [7] | H. Kawarada, On solutions of initial-boundary problem for \(u_t=u_{x x}+1/(1-u)\),, Publ. Res. Inst. Math. Sci., 10, 729 (1975) · Zbl 0306.35059 · doi:10.2977/prims/1195191889 |
| [8] | P. Lin, Quenching time optimal control for some ordinary differential equations,, preprint · Zbl 1437.49031 |
| [9] | P. Lin, Blowup time optimal control for ordinary differential equations,, SIAM J. Control Optim., 49, 73 (2011) · Zbl 1230.49001 · doi:10.1137/090764232 |
| [10] | J. Warga, <em>Optimal Control of Differential and Functional Equations</em>,, Academic Press (1972) · Zbl 0253.49001 |
| [11] | B. Yordanov, Finite-time blowup for wave equations with a potential,, SIAM J. Math. Anal., 36, 1426 (2005) · Zbl 1074.35075 · doi:10.1137/S0036141004440198 |
| [12] | Z. Zhang, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,, Nonlinear Anal., 72, 4594 (2010) · Zbl 1189.35033 · doi:10.1016/j.na.2010.02.036 |
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