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On \(\varepsilon\)-optimal controls for state constraint problems. (English) Zbl 0969.49019
As the authors state in the Introduction, the aim of this paper is to extend to state constrained problems the method in [F. H. Clarke, Yu. S. Ledyaev, E. D. Sontag and A. I. Subbotin, IEEE Trans. Aut. Contr. 42, No. 10, 1394-1407 (1997; Zbl 0892.93053)] to prove the existence of sub-optimal feedback controls for problems of the form: \[ u(x):=\displaystyle \inf_{\alpha(.)}J(x,\alpha(.)), \;J(x,\alpha(.)):= \int_0^\infty e^{-t} f(X(t;x,\alpha(.)),\alpha(t)) dt, \;x\in \overline{\Omega} \subset R^N, \] subject to: \[ X'(t)=g(X(t),\alpha(t)), \;X(0)=x, \quad X(t):=X(t;x,\alpha(.))\in \overline{ \Omega} \;\forall \;t\geq 0 \] where \(\Omega \subset R^N\) is a bounded open subset, \(\overline{\Omega}:= \text{Cl}(\Omega) \) and \(\alpha (.):[0,\infty)\to A\subset R^m\) are measurable controls.
The main result of the paper is Theorem 5.1 stating, essentially, that under some (rather restrictive) hypotheses on the boundary \(\partial \Omega\), for each \(\varepsilon >0\) there exist \(\widehat {\tau}>0\) and a mapping \(\widehat{\alpha}_\varepsilon (.):\overline{\Omega}\to A\) that is an \(\varepsilon\) -suboptimal feedback control in the following sense: for any \(x\in \overline{\Omega}\) the “piecewise constant” mapping \(\alpha_{\varepsilon ,x}(.)\) defined by: \[ \alpha_{\varepsilon ,x}(t):= \widehat{\alpha}_\varepsilon (x_k) \text{ if }t\in [k\widehat{\tau},(k+1) \widehat{\tau}), \;x_0=x, \;x_{k+1}=X(\widehat {\tau};x_k,\widehat{\alpha}_\varepsilon(x_k)), \;k=0,1,2,\dots \] is an \(\varepsilon\)-optimal control (with respect to the initial point \(x\)) in the sense that: \[ u(x)\leq J(x,\alpha_{\varepsilon ,x}(.))<u(x)+\varepsilon, \] where \(u(.)\) is the value function of the problem.
The proof of the main result takes some 4 pages and is based on a large number of very technical auxiliary results. Related results, in the slightly different context of “proximal analysis” may be found in [F. Clarke, Yu. S. Ledyaev and A. I. Subbotin, Tr. Mat. Inst. Steklova 224, 165-186 (1999; Zbl 0965.49022)].

MSC:
49N35 Optimal feedback synthesis
49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J52 Nonsmooth analysis
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