Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems with fractional discount rate

TL;DR

This paper derives the Hamilton-Jacobi-Bellman equation for nonlinear optimal control problems with fractional discount rates modeled by Mittag-Leffler functions, extending classical control theory to fractional calculus.

Contribution

It introduces a novel derivation of the HJB equation incorporating fractional discount rates described by Mittag-Leffler functions, expanding the framework of optimal control.

Findings

Derived HJB equation with fractional discount rate

Extended control theory to fractional calculus

Potential applications in systems with memory effects

Abstract

This paper derives the Hamilton-Jacobi-Bellman equation of nonlinear optimal control problems for cost functions with fractional discount rate from the Bellman's principle of optimality. The fractional discount rate is described by Mittag-Leffler function that can be considered as a generalized exponential function.