Abstract
In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.
Highlights
In the particular case where the system is given by x = F (x) and the function is B : Rn → R, this problem consists in establishing necessary and sufficient conditions such that the scalar function t → B(φ(t)) is nonincreasing for every solution t → φ(t) to x = F (x)
– When the scalar function B is lower semicontinuous (i.e., for each x ∈ Rn and for each sequence {xn}∞ n=0 ⊂ Rn with lim infn→∞ xn = x ∈ Rn, we have lim infn→∞ B(xn) ≥ B(x)) we transform the problem of showing that B is nonincreasing along the solutions to Hf = (C, F ) into characterizing forward pre-invariance of the set epiB ∩ (cl(C) × R), where epiB := {(x, r) ∈ Rn × R : r ≥ B(x)} is the epigraph of B and cl(C) is the closure of C, for the augmented constrained system xr
This paper characterizes the nonincrease of scalar functions along solutions to differential inclusions defined on a constrained set
Summary
Initial solutions to this problem deal with the particular case where both F and B are sufficiently smooth In such a basic setting, a necessary and sufficient condition for B to be nonincreasing is that the scalar product between the gradient of B and F is nonpositive at each x ∈ Rn; namely, ∇B(x), F (x) ≤ 0 for all x ∈ Rn. When B is not continuously differentiable, the problem requires nonsmooth analysis tools since the gradient of B may not be defined according to the classical sense. These extensions allow to cover general scenarios where F is a general set-valued map, and the system is a differential inclusion of the form x ∈ F (x), and B is merely continuous, or just semicontinuous In those extensions, the classical gradient ∇B is replaced by its nonsmooth versions, such as the proximal subderivative [8], denoted ∂P B, and the Clarke generalized gradient [5, 9], denoted ∂CB
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