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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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