## Universal feedback control via proximal aiming in problems of control under disturbance and differential games.(English. Russian original)Zbl 0965.49022

Proc. Steklov Inst. Math. 224, 149-168 (1999); translation from Tr. Mat. Inst. Steklova. 224, 165-186 (1999).
As the authors state in the Introduction, this article is a variant (in Russian !) of their Preprint: “Universal feedback via proximal aiming in control problems under disturbances and differential games” [Rapport CRM-2386, Univ. Montréal, 22 p. (1994)], concerning the construction of a discontinuous “universal feedback control” $$u_f(.,.)$$ that “approximately” minimizes the functional: $\Lambda_u(u_f)(t_0,x_0):=\sup_{x(.),v(.)}\Lambda (x(.),u(.),v(.)) (t_0,x_0),$
$\Lambda (x(.),u(.),v(.))(t_0,x_0):=l(x(T))-\int_{t_0}^T L(x(t), u(t),v(t)) dt$ subject to: $x'=f(x,u_f(t,x),v(t)), \;x(t_0)=x_0, \;u(t):=u_f(t,x(t))\in P\subset R^p, \;v(t)\in Q\subset R^q$ which is obviously related to the theory of differential games.
For the construction of an universal feedback control $$u_f(.,.)$$ the authors use the “upper Hamiltonian” and the corresponding marginal multifunction: $H^+(x,p):= \min_{u\in P}\max_{v\in Q} {\mathcal H}(x,p,u,v), \;{\mathcal H}(x,p,u,v):= \langle p,f(x,u,v)\rangle -L(x,u,v),$
$\widehat {U}^+(x,p):=\displaystyle {\text{Argmin}_{u\in P}} \{\max_{v\in Q} {\mathcal H}(x,p,u,v)\}$ and a continuous “proximal supersolution” $$\phi (.,.)$$ whose “proximal subgradients” in $$\partial_P\phi(t,x)$$, satisfy: $\zeta_t+H^+(x,\zeta_x)\leq 0 \;\forall (\zeta_t,\zeta_x) \in \partial_P\phi (t,x), \;(t,x)\in (-\infty ,T] \times R^n, \;\phi (T,x)\equiv l(x).$ The first main result of the paper is Theorem 4.1 stating, essentially, that under some hypotheses on the data, for any compact subset $$D_0\subset (-\infty ,T]\times R^n$$ there exists $$\varepsilon >0$$ such that certain “approximate selectors” $$u^\varepsilon_f(t,x)\in \widehat U^+(t,x,\zeta_ x^\varepsilon), \;(\zeta_t^\varepsilon,\zeta_x^\varepsilon)\in \partial_P\phi (t^\varepsilon , x^\varepsilon), \;(t,x)\in D_0$$ are “suboptimal feedback controls” in the sense that: $\Lambda_u(u_f^\varepsilon)(t_0,x_0)\leq \phi (t_0,x_0)+\varepsilon \;\forall \;(t_0,x_0)\in D_0.$ In the last section of the paper an analogous but much more complicated construction is described for proximal supersolutions that are only lower semicontinuous.
The authors mention the fact that this construction procedure has already been used in their article [F. H. Clarke, Yu. S. Ledyaev, E. D. Sontag and A. I. Subbotin, IEEE Trans. Autom. Control 42, No. 10, 1394-1407 (1997; Zbl 0892.93053)] to prove that “asymptotic controllability implies feedback stabilization”.
For the entire collection see [Zbl 0942.00074].

### MSC:

 49N35 Optimal feedback synthesis 49N70 Differential games and control 49L20 Dynamic programming in optimal control and differential games 49J52 Nonsmooth analysis 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games

Zbl 0892.93053