## Filippov’s and Filippov-Ważewski’s theorems on closed domains.(English)Zbl 0956.34012

The authors extend the known Filippov and Filippov-Ważewski theorems for differential inclusions to the case when the state variable $$x$$ is constrained to the possibly nonsmooth closure of an open subset in $$\mathbb{R}^n$$. They consider Cauchy problems of the form $x'\in F(t, x),\quad x(t_0)= x_1,\quad x(t)\in\overline\Theta,\tag{1}$ where $$\Theta\subset \mathbb{R}^n$$ is an open subset and, in contrast to other authors, a time interval is unbounded, here. More precisely, the authors extend the following statements:
1. There exists a constant $$L>0$$ such that, for any pair of initial conditions $$x_1,x_2\in \mathbb{R}^n$$ and any solution $$x_1(\cdot)$$ to $x'\in F(t,x),\quad x(t_0)= x_1,\tag{2}$ defined on $$[t_0, T]$$, there is a solution $$x_2(\cdot)$$ to (2) with $$x_1$$ replaced by $$x_2$$ such that $\sup_{t\in [t_0,T]}|x_1(t)- x_2(t)|+ \int^T_{t_0}|x_1'(t)- x_2'(t)|dt\leq L|x_1- x_2|.$ 2. For any trajectory $$x(\cdot)$$ to (2) with $$F(t,x)$$ replaced by $$\overline{\text{co}} F(t,x)$$ defined on $$[t_0, T]$$ and for any $$\varepsilon> 0$$, there exists a trajectory $$x_\varepsilon(\cdot)$$ to (2) such that $\sup_{t\in [t_0,T]}|x(t)- x_\varepsilon(t)|< \varepsilon.$ Let us note that in the case when $$\Theta$$ is nonsmooth just a $$C^0$$ estimate is proved in the first type assertion. Applications to the study of regularity of value functions of optimal control problems with state constraints are discussed, too.

### MSC:

 34A60 Ordinary differential inclusions 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 34H05 Control problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations
Full Text:

### References:

 [1] Arisawa, M.; Lions, P.-L., Continuity of admissible trajectories for state constraints control problems, Discrete contin. dynam. systems, 2, 297-305, (1996) · Zbl 0953.49004 [2] Aubin, J.P.; Cellina, A., Differential inclusions, (1984), Springer-Verlag [3] Aubin, J.P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser [4] Brezis, H., Analyse fonctionelle, (1982), Masson [5] Cannarsa, P.; Gozzi, F.; Soner, H.M., A boundary-value problem for hamilton – jacobi equations in Hilbert spaces, Appl. math. optim., 24, 197-220, (1991) · Zbl 0745.49025 [6] Capuzzo Dolcetta, I.; Lions, P.-L., Hamilton – jacobi equations and state constrained problems, Trans. amer. math. soc., 318, 643-668, (1990) · Zbl 0702.49019 [7] Crandall, M.G.; Lions, P.L., Viscosity solutions of hamilton – jacobi equations, Trans. amer. math. soc., 277, 1-42, (1983) · Zbl 0599.35024 [8] Forcellini, F.; Rampazzo, F., On nonconvex differential inclusions whose state is constrained in the closure of an open set, applications to dynamic programming, Differential integral equations, 12, 471-497, (1999) · Zbl 1015.34006 [9] Frankowska, H.; Rampazzo, F., Relaxation of control systems under state constraints, SIAM J. control, 37, 1291-1309, (1999) · Zbl 0924.34059 [10] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer Berlin · Zbl 0691.35001 [11] Ishii, H.; Koike, S., A new formulation of state constraints problems for first order pdes, SIAM J. control optim., 365, 554-576, (1996) · Zbl 0847.49025 [12] Loreti, P., Some properties of constrained viscosity solutions of hamilton – jacobi – bellman equations, SIAM J. control optim., 25, 1244-1252, (1987) · Zbl 0679.49036 [13] Loreti, P.; Tessitore, M.E., Approximation and regularity results on constrained viscosity solutions of hamilton – jacobi – bellman equations, J. math. systems estim. control, 4, 467-483, (1994) · Zbl 0830.49020 [14] Motta, M., On nonlinear optimal control problems with state constraints, SIAM J. control optim., 33, 1411-1424, (1995) · Zbl 0861.49018 [15] M. Motta, and, F. Rampazzo, Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints, preprint, 1997. · Zbl 0961.34003 [16] F. Rampazzo, and, R. Vinter, Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints, preprint, 1998. [17] Rockafellar, R.T., Clarke’s tangent cones and the boundaries of closed sets, Nonlinear anal., 145-154, (1979) · Zbl 0443.26010 [18] Soner, H.M., Optimal control with state-space constraints, SIAM J. control optim., 24, 552-561, (1986) · Zbl 0597.49023 [19] P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi-Bellman equations. II. Equations of control problems with state constraints, Adv. Differential Equations, to appear. · Zbl 1007.49016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.