Nonlinear operators and differential equations in Banach spaces
Nonlinear operators and differential equations in Banach spaces
Viability theory
Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations
SIAM Journal on Control and Optimization
Weak Tangency, Weak Invariance, and Carathéodory Mappings
Journal of Dynamical and Control Systems
Journal of Dynamical and Control Systems
Approximate weak invariance for differential inclusions in Banach spaces
Journal of Dynamical and Control Systems
Nonlinear evolution inclusions with one-sided perron right-hand side
Journal of Dynamical and Control Systems
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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 ε-trajectory corresponding to a differential inclusion.