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Stabilization of uncertain control systems through piecewise constant feedback. (English) Zbl 0891.93068
An uncertain system with dynamics defined by $\dot{x}(t)=f(x(t),u(t),d(t)),\quad x(t) \in R^n,\;u(t) \in U \subset R^m,\;d(t) \in D \subset R^l,$ where $$t$$, $$u$$, $$d$$, and $$x$$ are the time, the control, the disturbance and the state, is considered. The sets $$U$$ and $$D$$ are supposed to be closed. Let $$V:R^n \to R^+$$ be a given function and $$a$$ be a positive parameter. The following approximate stabilization problem is studied: Fix $$T>0$$ and $$\eta>0$$. The aim is to find a control law $$u=u(t,x)\in U$$ such that for any measurable $$d(.)$$ the system of ordinary differential equations $\dot{x}(t)=f(x(t),u(t,x(t)),d(t)),\;t \in [0,T]$ has at least one solution and any solution $$x(.)$$ satisfies $V(x(t))\leq V(x(0))e^{-at}+\eta,\;t \in [0,T].$ Assuming that $$V$$ is Lipschitz continuous and satisfies $\inf_{u \in U} \sup_{d\in D} \;\langle f(x,u,d),p\rangle \leq aqV(x)$ for each proximal normal $$(p,q)$$ to the epigraph $$\text{epi}V$$ of $$V$$ at the point $$(x,V(x))$$, the existence of a piecewise constant feedback satisfying the approximate stabilization property is proved under natural technical assumptions on $$f$$. This is the main result of the paper. Its proof relies on the fact that the conditions imposed on $$V$$ imply viability of the set $$\text{epi}V$$ with respect to the following augmented system: $\dot{x}(t)=f(x(t),u(t),d(t)),\;u(t) \in U,\;d(t) \in D,\;\dot{w}(t)=-aw(t).$ Thus, for any $$\varepsilon > 0$$ there exists a piecewise control such that for any measurable disturbance $$d(.)$$ the corresponding trajectory $$(x(.),w(.))$$ starting from $$(x_0, V(x_0))$$ is defined on $$[0,T]$$ and the distance between $$(x(t),w(t)), \;t \in [0,T]$$ and $$\text{epi}V$$ is not greater than $$\varepsilon$$.

##### MSC:
 93D21 Adaptive or robust stabilization 93D15 Stabilization of systems by feedback 93C41 Control/observation systems with incomplete information
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##### References:
 [1] Aubin, J.-P., Viability theory, (1992), Birkhäuser Basel [2] P. Cardaliaguet, Domaines Discriminants en Jeux Différentiels, University Paris IX Dauphine, 1993 [3] Cardaliaguet, P., A differential game with two players and one target, SIAM J. control optim., 34, 1441-1460, (1996) · Zbl 0853.90136 [4] Corless, M.; Leitmann, G., Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE trans. automat. control, 26, (1981) · Zbl 0473.93056 [5] Filippov, A.F., Differential equations with discontinuous right hand sides, (1988), Kluwer Academic Dordrecht · Zbl 0664.34001 [6] H. Frankowska, M. Quincampoix, Dissipative control systems and disturbance attenuation for nonlinearH∞, Appl. Math. Optim. · Zbl 0943.93022 [7] Gutman, S., Uncertain dynamical systems: A Lyapunov MIN-MAX approach, IEEE trans. automat. control, 24, 437-443, (1979) · Zbl 0416.93076 [8] S. Gutman, G. Leitmann, Stabilizing feedback control for dynamical systems with bounded uncertainty, in, Proceedings of the IEEE Conference on Decision and Control, Clearwater, Florida, 1976, 94, 99 · Zbl 0334.93030 [9] Kokotovic, P.V.; Freeman, R.A., Robust nonlinear control design, (1996), Birkhäuser Basel [10] Krasovskii, N.N.; Subottin, A.I., Game-theoretical control problems, (1988), Springer-Verlag New York/Berlin [11] Leitmann, G., Guaranteed asymptotic stability for a class of uncertain linear dynamical systems, J. optim. theory appl., 27, (1979) · Zbl 0377.93064 [12] Quincampoix, M., Differential inclusions and target problems, SIAM J. control optim., 30, 324-335, (1992) · Zbl 0862.49006 [13] Seube, N., Robust stabilization of uncertain systems, J. math. anal. appl., 196, 452-466, (1995) · Zbl 0847.93054
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