Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Operator-Theoretic Framework for the Optimal Control Problem of Nonlinear Caputo Fractional Systems

Dev Prakash Jha, Raju K. George

TL;DR

The paper develops an operator-theoretic framework for the optimal control of nonlinear Caputo fractional systems with nonlocal initial data by reformulating the dynamics as a Hammerstein-type integral equation and introducing the $\alpha$-order solution operator $T_{\alpha}$ and resolvent family $S_{\alpha}$. It proves existence (and, in certain cases, uniqueness) of mild solutions and of optimal control–state pairs under a range of monotonicity, compactness, and lower semicontinuity assumptions, and derives an explicit quadratic-cost optimality system via Fr\u00e9chet differentiability with adjoint relations. The results connect to Pontryagin-type principles and are extended to systems governed by differential equations, including parabolic PDEs, with both spatial discretization (Galerkin) and temporal discretization (Caputo-L1, backward Euler). An MOA-based numerical scheme is developed to compute discrete optimal controls, and a 1D fractional diffusion example demonstrates the method's efficacy and convergence. Overall, the work provides a comprehensive, verifiable framework for analysis and computation of optimal control problems for Caputo fractional systems with nonlocal initial conditions, spanning abstract Hammerstein theory to concrete PDE applications.

Abstract

This paper addresses the optimal control problem for a class of nonlinear fractional systems involving Caputo derivatives and nonlocal initial conditions. The system is reformulated as an abstract Hammerstein-type operator equation, enabling the application of operator-theoretic techniques. Sufficient conditions are established to guarantee the existence of mild solutions and optimal control-state pairs. The analysis covers both convex and non-convex scenarios through various sets of assumptions on the involved operators. An optimality system is derived for quadratic cost functionals using the Gâteaux derivative, and the connection with Pontryagin-type minimum principles is discussed. Illustrative examples demonstrate the effectiveness of the proposed theoretical framework.

An Operator-Theoretic Framework for the Optimal Control Problem of Nonlinear Caputo Fractional Systems

TL;DR

The paper develops an operator-theoretic framework for the optimal control of nonlinear Caputo fractional systems with nonlocal initial data by reformulating the dynamics as a Hammerstein-type integral equation and introducing the -order solution operator and resolvent family . It proves existence (and, in certain cases, uniqueness) of mild solutions and of optimal control–state pairs under a range of monotonicity, compactness, and lower semicontinuity assumptions, and derives an explicit quadratic-cost optimality system via Fr\u00e9chet differentiability with adjoint relations. The results connect to Pontryagin-type principles and are extended to systems governed by differential equations, including parabolic PDEs, with both spatial discretization (Galerkin) and temporal discretization (Caputo-L1, backward Euler). An MOA-based numerical scheme is developed to compute discrete optimal controls, and a 1D fractional diffusion example demonstrates the method's efficacy and convergence. Overall, the work provides a comprehensive, verifiable framework for analysis and computation of optimal control problems for Caputo fractional systems with nonlocal initial conditions, spanning abstract Hammerstein theory to concrete PDE applications.

Abstract

This paper addresses the optimal control problem for a class of nonlinear fractional systems involving Caputo derivatives and nonlocal initial conditions. The system is reformulated as an abstract Hammerstein-type operator equation, enabling the application of operator-theoretic techniques. Sufficient conditions are established to guarantee the existence of mild solutions and optimal control-state pairs. The analysis covers both convex and non-convex scenarios through various sets of assumptions on the involved operators. An optimality system is derived for quadratic cost functionals using the Gâteaux derivative, and the connection with Pontryagin-type minimum principles is discussed. Illustrative examples demonstrate the effectiveness of the proposed theoretical framework.

Paper Structure

This paper contains 7 sections, 18 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis of the Existence of an Optimal Pair in the Abstract Framework
  4. OPTIMALITY SYSTEM FOR THE ABSTRACT PROBLEM
  5. Optimality Results for Systems Governed by Differential Equations
  6. Application
  7. Numerical experiment

Key Result

Lemma 2.1

Suppose that $A \in \mathcal{A}^{\alpha}(\omega_{0}, \theta_{0})$, and that both the $\alpha$-order solution operator $\{T_{\alpha}(t)\}_{t \geq 0}$ and the $\alpha$-order resolvent operator $\{S_{\alpha}(t)\}_{t \geq 0}$ are compact. Then, for every $t > 0$, the following properties hold:

Figures (2)

  • Figure 1: Profile of exact solution and control function
  • Figure 2: Profile of approximate solution and optimal control

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 31 more