Duality And Optimality Conditions In Non-Differentiable Control Problems

Rajesh Kumar

doi:10.71097/IJSAT.v17.i2.11267

Duality And Optimality Conditions In Non-Differentiable Control Problems

Author(s)	Dr. Rajesh Kumar
Country	India
Abstract	Optimal control theory has long served as a cornerstone of modern applied mathematics, providing powerful tools for the analysis and regulation of dynamic systems encountered in engineering, economics, management sciences, aerospace technology, and industrial optimization. Most classical formulations of optimal control rely on the assumption that both the system dynamics and the objective functional possess sufficient smoothness properties. In many practical situations, however, these assumptions fail to hold. Real-world systems frequently involve non-differentiable elements such as absolute value functions, switching costs, threshold effects, frictional forces, and sparsity-inducing penalties. The presence of such non-smooth structures complicates the analytical treatment of control problems and requires the use of generalized mathematical techniques. Among the most effective approaches for studying these problems is duality theory, which provides alternative representations of optimization models and establishes fundamental relationships between primal and dual formulations. Through these relationships, one can derive bounds on optimal values, characterize optimal solutions, and obtain meaningful theoretical insights even when conventional differentiability assumptions are absent. This paper examines duality and optimality conditions for a class of non-differentiable control problems within a convex analytical framework. Basic concepts and mathematical preliminaries are developed using generalized gradients and non-smooth analysis. Weak duality, strong duality, and converse duality results are formulated and proved rigorously. Furthermore, necessary and sufficient optimality conditions are established through generalized Hamiltonian techniques. The findings demonstrate that duality principles remain valid in non-smooth settings and continue to provide an effective mechanism for analyzing complex control systems. The theoretical framework developed herein contributes to the broader literature on non-smooth optimization and offers a foundation for future analytical and computational investigations.
Keywords	Optimal control, non-differentiable optimization, duality theory, generalized gradient, convex analysis, non-smooth systems, Hamiltonian methods, optimality conditions.
Field	Mathematics
Published In	Volume 17, Issue 2, April-June 2026
Published On	2026-06-13
DOI	https://doi.org/10.71097/IJSAT.v17.i2.11267

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Duality And Optimality Conditions In Non-Differentiable Control Problems

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