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Strong invariance and one-sided Lipschitz multifunctions. (English) Zbl 1068.34010
The paper is devoted to the study of some viability issues concerning differential inclusions in the Euclidean space $\bbfR^n$. The authors study the Cauchy problem $$x'\in F(t, x), \quad x\in S, \quad t\in I= [t_0, t_1), \quad x(t_0)= x_0\in S,\tag *$$ where $S\subset\bbfR^n$ is closed, $F$ has convex compact values, is almost upper semicontinuous and integrably bounded. In order to establish the strong invariance (strong viability), that is to show that any solution to $(*)$ with $x_0\in S$ remains in $S$, they introduce the max-Hamiltonian $H_F(t,x,y)= \sup_{z\in F(t,x)}\langle z, y\rangle$ and, instead of the Lipschitz continuity of $F$ which has been usually assumed to get the strong invariance, they assume that $H_F$ satisfies the so-called one sided Lipschitz estimates of the form $H_F(t,x,x- y)- H_F(t,y,x- y)\le k(t)\Vert x- y\Vert$ where $k$ is locally integrable. They further prove that, under these assumptions, $S$ is strongly invariant, provided that the for almost all $t$, asymptotically $H(t,x,y)\le 0$ for all vectors $y$ in the proximal normal cone $N^P_S(x)$ at $x\in S$. The authors study also the situation of time dependent state constraints $S(t)$, $t\in I$, and show conditions equivalent to the weak and strong invariance. These conditions are stated in terms of Hamilton-Jacobi-type inequalities.

##### MSC:
 34A60 Differential inclusions 49J53 Set-valued and variational analysis 49L99 Hamilton-Jacobi theories, including dynamic programming
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##### References:
 [1] Aubin, J. -P.: Viability theory. (1991) · Zbl 0755.93003 [2] Aubin, J. -P.; Cellina, A.: Differential inclusions. (1984) · Zbl 0538.34007 [3] Bony, J. -M.: Principe du maximun. Inégalité de Harnack et unicité du problème de Cauchy pour LES opérateurs elliptiques dégénérés, ann. Inst. Fourier Grenoble 19, 277-304 (1969) [4] D. Bothe, Multivalued differential equations on graphs and applications, Ph.D. Thesis, Paderborn, 1992. · Zbl 0789.34013 [5] Brezis, H.: On a characterization of flow-invariant sets. Commun. pure appl. Math. 23, 261-263 (1970) · Zbl 0191.38703 [6] Clarke, F.: Generalized gradients and applications. Trans. am. Math. soc. 205, 247-262 (1975) · Zbl 0307.26012 [7] Clarke, F.; Ledyaev, Yu.; Radulescu, M.: Approximate invariance and differential inclusions in Hilbert spaces. J. dyn. Control syst. 3, 493-518 (1997) · Zbl 0951.49007 [8] Clarke, F.; Ledyaev, Yu.; Stern, R.; Wolenski, P.: Nonsmooth analysis and control theory. (1998) · Zbl 1047.49500 [9] Crandall, M. G.: A generalization of Peano’s theorem and flow invariance. Proc. am. Math. soc. 36, 151-155 (1972) · Zbl 0271.34084 [10] Deimling, K.: Multivalued differential equations. (1992) · Zbl 0760.34002 [11] Donchev, T.: Functional differential inclusions with monotone right-hand side. Nonlinear anal. 16, 543-552 (1991) [12] Donchev, T.: Properties of the reachable set of control systems. Syst. control lett. 46, 379-386 (2002) · Zbl 1003.93003 [13] Donchev, T.; Ríos, V.; Wolenski, P.: A characterization of strong invariance for perturbed dissipative systems, optimal control, stabilization, and nonsmooth analysis. Lecture notes in control and information sciences 301, 343-349 (2004) [14] T. Donchev, V. Ríos, P. Wolenski, Approximate semi-solutions to Hamilton -- Jacobi equations, submitted. [15] Frankowska, H.; Plaskacz, S.; Rzezuchowski, T.: Measurable viability theorems and the Hamilton -- Jacobi -- Bellman equation. J. differ. Equations 116, 265-305 (1995) · Zbl 0836.34016 [16] Frankowska, H.; Plaskacz, S.: A measurable upper semicontinuous viability theorem for tubes. Nonlinear anal. Theory methods appl. 26, 565-582 (1996) · Zbl 0838.34017 [17] Bhaskar, T. Gnana; Lakshmikantham, V.: Generalized proximal normals and flow invariance for differential inclusions. Nonlinear stud. 10, 29-37 (2003) · Zbl 1035.34009 [18] Haddad, G.: Monotone trajectories of differential inclusions and functional differential inclusions with memory. Isr. J. Math. 39, 83-100 (1981) · Zbl 0462.34048 [19] Hartman, P.: On invariant sets and on a theorem of wazewski. Proc. am. Math. soc. 32, 511-520 (1972) · Zbl 0272.34049 [20] Krastanov, M.: Forward invariant sets, homogeneity and small-time local controllability. Geometry in nonlinear control and differential inclusions, Banach center publications 32, 287-300 (1995) · Zbl 0839.93010 [21] Lade, G.; Lakshmikantham, V.: On flow invariant sets. Pacific J. Math. 51, 215-220 (1974) · Zbl 0251.34033 [22] Ledyaev, Yu.S.: Criteria for viability of trajectories of nonautonomous differential inclusions and their applications. J. math. Anal. appl. 182, No. 1, 165-188 (1994) · Zbl 0798.34019 [23] Nagumo, M.: Uber die lage der integralkurven gewöhnlicher differentialgleichungen. Proc. phys. Math. soc. Japan 24, 551-559 (1942) · Zbl 0061.17204 [24] Rapaport, A. E.; Vinter, R. B.: Invariance properties of time measurable differential inclusions and dynamic programming. J. dyn. Control syst. 2, No. 3, 423-448 (1996) · Zbl 0943.34011 [25] Redheffer, R. M.; Walter, W.: Flow-invariant sets and differential inequalities in normed spaces. Appl. anal. 5, 149-161 (1975) · Zbl 0353.34067 [26] Ríos, V.; Wolenski, P.: A characterization of strongly invariant systems for a class of non-Lipschitz multifunctions. Proceedings of the 42th IEEE conference on decision and control, maui, hawaii USA, 2593-2594 (2003) [27] Tolstonogov, A.: Differential inclusions in a Banach space. (2000) · Zbl 1021.34002 [28] Veliov, V. M.: Sufficient conditions for viability under imperfect measurement. Set-valued anal. 1, 305-317 (1993) · Zbl 0802.49029