[Paper Review] Approximate controllability and optimal control of impulsive fractional semilinear delay differential equations with non-local conditions

Lakshman Mahto, Syed Abbas|arXiv (Cornell University)|Apr 11, 2013

Nonlinear Differential Equations Analysis11 references4 citations

TL;DR

This paper establishes approximate controllability and optimal control for impulsive fractional semilinear delay differential equations with non-local conditions using Sadovskii’s fixed point theorem, semigroup theory, and direct minimization of a cost functional. The key contribution is proving the existence of mild solutions and optimal controls under mild Lipschitz and compactness conditions, validated via a concrete example in a Hilbert space setting.

ABSTRACT

In this paper we study the approximate controllability and existence of optimal control of impulsive fractional semilinear delay differential equations with non-local conditions. We use Sadovskii's fixed point theorem, semigroup theory of linear operators and direct method for minimizing a functional to establish our results. At the end we give an example to illustrate our analytical findings.

Motivation & Objective

• To investigate the approximate controllability of impulsive fractional semilinear delay differential equations with non-local initial conditions.
• To establish the existence of optimal control for such systems via direct minimization of a cost functional.
• To extend existing results on controllability and optimal control to fractional-order systems with impulses, delays, and non-local conditions.
• To provide a theoretical framework applicable to real-world systems in engineering and physics with memory and abrupt changes.

Proposed method

• Utilizes Sadovskii’s fixed point theorem to prove the existence of mild solutions for the impulsive fractional delay system.
• Applies semigroup theory of sectorial linear operators to handle the fractional derivative term $ D_t^eta $ with $ \beta \in (1,2) $.
• Employs a direct method to minimize a cost functional $ J(u) $, ensuring existence of optimal control under lower semi-continuity and convexity assumptions.
• Establishes convergence of control sequences via norm estimates involving the resolvent operator $ S_\alpha(t) $ and Lipschitz conditions on nonlinearity and impulses.
• Uses the Baldrac theorem to confirm that the infimum of the cost functional is attained at an optimal control.
• Validates theoretical results with a concrete example involving a fractional partial differential equation on $ L^2(0,\pi) $.

Experimental results

Research questions

• RQ1 Under what conditions is the mild solution of an impulsive fractional semilinear delay system with non-local conditions approximately controllable?
• RQ2 Can an optimal control be guaranteed for such systems when the cost functional is lower semi-continuous and convex?
• RQ3 How do impulses, delays, and non-local initial conditions affect the controllability and optimal control of fractional-order systems?
• RQ4 What role does the sectorial operator and analytic semigroup play in ensuring existence and regularity of solutions?

Key findings

• The system (1.1) admits a mild solution under mild Lipschitz and compactness conditions on the nonlinear terms and impulses.
• Approximate controllability is achieved when the operator $ A $ is sectorial and the control-to-state map is compact, ensuring dense image of reachable states.
• The optimal control $ u^0 \in U_{ad} $ exists and minimizes the cost functional $ J(u) $, with $ J(u^0) \leq J(u) $ for all admissible controls.
• Convergence of control sequences $ u^m \to u^0 $ implies strong convergence of corresponding state trajectories $ x^m \to x^0 $, ensuring stability of the optimal solution.
• The example in Section 5 demonstrates that a fractional PDE with delay and impulses can be reformulated as an abstract system satisfying all assumptions of Theorem 3.4 and Theorem 4.1.
• The performance index $ J(u) $ is finite and attains its minimum, confirming the existence of an optimal control in $ L^2(I,U) $.