## 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

### Citations:

Zbl 0982.93053; Zbl 0965.49022; Zbl 0892.93053
Full Text:

### References:

 [1] Bardi, M; Capuzzo Dolcetta, I, Optimal control and viscosity solutions of hamilton – jacobi – bellman equations, (1997), Birkhäuser · Zbl 0890.49011 [2] Barles, G, Solutions de viscosité des equations de hamilton – jacobi, (1994), Springer [3] Brezis, H, Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1973), North Holland Amsterdam · Zbl 0252.47055 [4] Clarke, F.H; Ledyaev, Y.S; Sontag, E.D; Subbotin, A.I, Asymptotic controllability implies feedback stabilization, IEEE trans. automat. control, Vol. 42, 1394-1407, (1997) · Zbl 0892.93053 [5] Ishii, H; Koike, S, A new formulation of state constraint problems for first order PDE’s, SIAM J. control optim., Vol. 36, 554-571, (1996) · Zbl 0847.49025 [6] Koike, S, On the state constraint problem for differential games, Indiana univ. math. J., Vol. 44, 467-487, (1995) · Zbl 0840.49016 [7] Loreti, P; Tessitore, M.E, Approximation and regularity results on constrained viscosity solution of hamilton – jacobi – bellman equations, J. math. systems estimation control, Vol. 4, 467-483, (1994) · Zbl 0830.49020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.