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DOI: 10.23952/cot.2027.6
Received August 5, 2025; Accepted February 16, 2026; Published online July 15, 2026
Abstract. In this paper, we present a brief review of optimal control theory for evolution equations on Banach spaces. We consider classical controls, relaxed controls, feedback controls and operator valued measures as controls. We consider general systems containing (non Lipschitz) continuous as well as measurable vector fields which admit measure valued solutions extending the standard notions of classical, strong, mild and weak solutions. We present existence of optimal control policies and develop necessary conditions of optimality whereby one can determine the optimal controls. Based on the necessary conditions of optimality, we present an algorithm including a proof of its convergence whereby the optimal policies can be constructed. Further, we consider non-convex control problems as special cases where the relaxed controls specialize to switching controls, generalizing the bang-bang principle.
How to Cite this Article:
N.U. Ahmed, A brief review of optimal control theory for systems governed by evolution equations on Banach spaces, Commun. Optim. Theory 2027 (2027) 6.