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The value function of singularly perturbed control systems. (English) Zbl 0958.49019
In this paper the authors significantly extend certain existing results concerning the asymptotic behaviour as $$\varepsilon \to 0$$ of the value function $V_\varepsilon (t,x,y)=\inf_{u(.)} \left\{\int_t^1L(x(\tau),y(\tau),u(\tau))d\tau +\psi (x(1))\right\}, \quad (t,x,y)\in [0,1)\times \mathbb{R}^m\times \mathbb{R}^n$ where the infimum is taken over all measurable (control) functions $$u(.):[t,1]\to U\subset \mathbb{R}^l$$ that “produce” the unique (absolutely continuous) solution $$(x(.),y(.))$$ of the problem: $x'(\tau)=f(x(\tau),y(\tau),u(\tau)), \;x(t)=x, \;\varepsilon y'(\tau)= g(x(\tau),y(\tau),u(\tau)), \;y(t)=y.$ Under some hypotheses on the value function itself one proves first that for any sequence $$\varepsilon_k \to 0$$ there exist a subsequence, say $$\varepsilon_j\to 0$$, and a “cluster function” $$V(.,.)$$, such that $$V_{\varepsilon_j}(t,x,y)\to V(t,x)$$ uniformly on compact subsets; next, the authors introduce the rather abstract “limit Hamiltonians”: $H_0(x,\lambda):=\lim_{s\to \infty}H(x,\lambda,s,y)$ $H(x,\lambda,s,y)=-\inf_{u(.)}\left\{ {{1}\over {s}}\int_0^s[ L(x,y(\tau), u(\tau)) +\lambda f(x,y(\tau),u(\tau))] d\tau \right\}$ where $$u(.)$$ are measurable control functions and $$y(.)$$ is the unique solution of the problem: $$y'(\tau)=g(x,y(\tau),u(\tau)), \;y(0)=y$$ and prove (on some 5 pages) their main result, Theorem 5.3, stating that under certain hypotheses on $$V\varepsilon(.,.,.)$$, $$H(.,.,.,.)$$, $$H_0(.,.)$$, any “cluster function” $$V(.,.)$$, of $$V_\varepsilon$$, is a viscosity solution of the (“limit”) Hamilton-Jacobi equation: $-{{\partial V}\over {\partial t}}+H_0(x, {{\partial V}\over {\partial x}})=0, \;V(1,x)=\psi (x).$ In Theorem 6.3 one identifies certain (more explicit) properties of the data that imply the rather implicit hypotheses of the main result and in a number of comments and examples the authors compare their results with previous work, in particular with those in [F. Bagagiolo and M. Bardi, SIAM J. Control Optimization 36, No. 6, 2040-2060 (1998; Zbl 0953.49031)] and [P.-L. Lions, “Generalized solutions of Hamilton-Jacobi equations” (1982; Zbl 0497.35001)], where problems “without order reduction hypothesis” are considered.

##### MSC:
 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 49L20 Dynamic programming in optimal control and differential games 34E15 Singular perturbations, general theory for ordinary differential equations
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