Applied Set-Valued Analysis and Optimization

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DOI: 10.23952/asvao.8.2026.3.07
Volume 8, Issue 3, 1 December 2026, Pages 393-419

 

Abstract. We investigate optimal control problems for abstract bilinear evolution systems in infinite-dimensional Hilbert spaces. Working within a Gelfand triple structure, we analyze the minimization of a tracking-type cost functional subject to a bilinear state equation. We establish well-posedness results for weak solutions under minimal regularity assumptions and prove existence of optimal controls via direct methods. We then derive first-order necessary conditions through differentiation of the control-to-state map, and develop a comprehensive second-order analysis yielding sufficient conditions for local optimality: we prove that controls satisfying first-order conditions and positive definiteness of the cost functional Hessian are strict local minimizers. Our analysis accounts for the two-norm discrepancy between cost functional differentiability and coercivity spaces. The theoretical results provide a rigorous foundation for numerical implementation, while their practical relevance is illustrated through an application to a coupled reaction-diffusion system modeling biological populations.

 

How to Cite this Article:
R. El Mezegueldy, Necessary and sufficient optimality conditions for a class of bilinear systems in infinite-dimensional spaces, Appl. Set-Valued Anal. Optim. 8 (2026), 393-419.