Abstract
Consider a mapping F from a Hilbert space H to the subsets of H, which is upper semicontinuous/Lipschitz, has nonconvex, noncompact values, and satisfies a linear growth condition. We give the first necessary and sufficient conditions in this general setting for a subset S of H to be approximately weakly/strongly invariant with respect to approximate solutions of the differential inclusion \(\dot x(t) \in F(x)\) The conditions are given in terms of the lower/upper Hamiltonians corresponding to F and involve nonsmooth analysis elements and techniques. The concept of approximate invariance generalizes the well-known concept of invariance and in turn relies on the notion of an ∈-trajectory corresponding to a differential inclusion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-P. Aubin, Viability theory. Birkhäuser, Boston, 1991.
J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, New York, 1984.
J.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier Grenoble 19 (1969), 277–304.
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Am. Math. Soc. 303 (1987), 517–527.
J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach space. Part I: Theory. Can. J. Math. 38 (1986), 431–452.
O. Cârja and I. I. Vrabie, Some new viability results for semilinear differential inclusions. Preprint, 1996.
F. H. Clarke, Generalized gradients and applications. Trans. Am. Math. Soc. 205 (1975), 247–262.
_____, Methods of dynamic and nonsmooth optimization. In: CBMSNSF Regional Conference series in Applied Mathematics, Vol. 57. SIAM, Philadelphia, 1989.
_____, Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, Vol. 5. SIAM, Philadelphia, 1990. (Originally published by Wiley Interscience, New York, 1983).
F. H. Clarke and J.-P. Aubin, Monotone invariant solutions to differential inclusions. J. London Math. Soc. 16 (1977), 357–366.
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Introduction to nonsmooth analysis, (textbook in preparation).
_____, Qualitative properties of trajectories of control systems: A survey. J. Dynam. and Control Syst. 1 (1995), 1–48.
F. H. Clarke, Yu. S. Ledyaev, and P. R. Wolenski. Proximal analysis and minimization principles. J. Math. Anal. Appl. 196 (1995), 722–735.
M. G. Crandall, A generalization of Peano's existence theorem and flow invariance. Proc. Am. Math. Soc. 36 (1972), 151–155.
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc. 277 (1983), 1–42.
K. Deimling, Ordinary differential equations in Banach spaces. Lect. Notes Math. 596 (1977).
_____, Multivalued differential equations, de Gruyter, Berlin, 1992.
J. Diestel and J. J. Uhl, Vector measures. Am. Math. Soc., Providence, 1977.
H. Frankowska, Lower semicontinuous solutions of the Hamilton-Jacobi equation. SIAM J. Control Optimiz. 31 (1993), 257–272.
Kh. Guseinov, A. I. Subbotin, and V. N. Ushakov, Derivatives for multivalued mappings with applications to game-theoretical problems of control. Probl. Control. Inf. Theory 14 (1985), 155–167.
Kh. Guseinov and V. N. Ushakov, Strongly and weakly invariant sets with respect to differential inclusions. (Russian) Differ. Uravneniya 26 (1990), 1888–1894.
N. N. Krasovskii and A. I. Subbotin, Positional differential games. (Russian) Nauka, Moscow, 1974. (Revised English translation: Game-theoretical control problems, Springer-Verlag, New York, 1988).
M. Krastanov, Ordinary forward invariant sets, homogeneity and smalltime local controllability. In: Banach Center Publications, Vol. 32. Polish Academy of Sciences, 1995.
P. D. Loewen, Optimal control via nonsmooth analysis. Vol. 2. CRM Proc. and Lect. Notes, Providence, Am. Math. Soc., 1993.
N. H. Pavel, Differential equations, flow invariance and applications. Vol. 113. Research Notes in Math., Pitman, 1984.
R. H. Jr. Martin, Nonlinear operators and differential equations in Banach spaces. Wiley, New York, 1976.
M. L. Radulescu and F. H. Clarke, Geometric approximation of proximal normals, (submitted).
R. Redheffer and W. Walter, A differential inequality for the distance function in normed linear spaces. Math. Ann. 211 (1974), 299–314.
Shuzhong Shi, Viability theorems of a class of differential-operator inclusions. J. Differ. Eqs. 79 (1989), 232–257.
A. I. Subbotin, A generalization of the basic equation of the theory of differential games. Sov. Math. Dokl. 22 (1980), 358–362.
_____, Generalized solutions of first-order PDEs: The dynamical optimization perspective. Birkhäuser, Boston, 1995.
V. M. Veliov, Sufficient conditions for viability under imperfect measurement. Set-Valued Anal. 1 (1993), 305–317.
J. A. Yorke, A continuous differential equation in Hilbert space without existence. Funkc. Ekvac. 13 (1970), 19–21.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Clarke, F.H., Ledyaev, Y.S. & Radulescu, M.L. Approximate Invariance and Differential Inclusions in Hilbert Spaces. Journal of Dynamical and Control Systems 3, 493–518 (1997). https://doi.org/10.1023/A:1021873607769
Issue Date:
DOI: https://doi.org/10.1023/A:1021873607769