- Equivalence of minimax and viscosity options of path-dependent Hamilton-Jacobi equations
Authors: Mikhail Gomoyunov, Anton Plaksin
Summary: Within the paper, we take into account a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the house of steady capabilities. Such equations come up from optimum management issues and differential video games for time-delay techniques. We examine generalized options of the thought of Hamilton-Jacobi equation each within the minimax and within the viscosity sense. A minimax resolution is outlined as a practical which epigraph and subgraph fulfill sure circumstances of weak invariance, whereas a viscosity resolution is outlined by way of a pair of inequalities for coinvariant sub- and super-gradients. We show that these two notions are equal, which is the primary results of the paper. As a corollary, we receive comparability and uniqueness outcomes for viscosity options of a Cauchy drawback for the thought of Hamilton-Jacobi equation and a right-end boundary situation. The proof is predicated on a sure property of the coinvariant subdifferential. To determine this property, we develop a way going again to the proofs of multidirectional mean-value inequalities. Particularly, the absence of the native compactness property of the underlying steady operate house is overcome by utilizing Borwein-Preiss variational precept with an acceptable guage-type practical
2. Minimax Options of Hamilton — Jacobi Equations with Fractional Coinvariant Derivatives
Authors: Mikhail Gomoyunov
Summary: We take into account a Cauchy drawback for a Hamilton — Jacobi equation with coinvariant derivatives of an order α∈(0,1). Such issues come up naturally in optimum management issues for dynamical techniques which evolution is described by bizarre differential equations with the Caputo fractional derivatives of the order α. We suggest a notion of a generalized within the minimax sense resolution of the thought of drawback. We show {that a} minimax resolution exists, is exclusive, and is in line with a classical resolution of this drawback. Particularly, we give a particular consideration to the proof of a comparability precept, which requires development of an appropriate Lyapunov — Krasovskii practical.
