The present paper studies a new class of problems of optimal control theory with linear second order self-adjoint Sturm-Liouville type differential operators and with functional and non-functional endpoint constraints. Sufficient conditions of optimality, containing both the second order Euler-Lagrange and Hamiltonian type inclusions are derived. The presence of functional constraints generates a special second order transversality inclusions and complementary slackness conditions peculiar to inequality constraints; this approach and results make a bridge between optimal control problem with Sturm-Liouville type differential differential inclusions and constrained mathematical programming problems in finite-dimensional spaces.The idea for obtaining optimality conditions is based on applying locally-adjoint mappings to Sturm-Liouville type set-valued mappings. The result generalizes to the problem with a second order non-self-adjoint differential operator. Furthermore, practical applications of these results are demonstrated by optimization of some semilinear optimal control problems for which the Pontryagin maximum condition is obtained. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

中文翻译：

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具有端点约束的Sturm-Liouville型演化微分包含的最优控制

本文研究了带有线性二阶自伴Sturm-Liouville型微分算子以及功能和非功能端点约束的最优控制理论的一类新问题。推导了同时包含二阶欧拉-拉格朗日和哈密顿型包含物的最优性的充分条件。功能约束的存在会产生特殊的二阶横向包含和不平等约束特有的互补松弛条件；这种方法和结果在带有Sturm-Liouville型微分微分包含的最优控制问题和有限维空间中的受限数学规划问题之间架起了桥梁。获得最优性条件的想法是基于将局部伴随映射应用于Sturm-Liouville类型集值映射。结果推广到二阶非自伴微分算子的问题。此外，这些结果的实际应用通过对一些具有Pontryagin极大条件的半线性最优控制问题的优化得到了证明。数值例子说明了所得理论结果的可行性和有效性。