Relaxation in nonconvex optimal control problems with subdifferential operators
Relaxation in nonconvex optimal control problems with subdifferential operators
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DOI：
10.1007/s10958-007-0021-9
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发表时间：
2007-02
影响因子：
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通讯作者：
A. Tolstonogov
中科院分区：
文献类型：
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作者：
A. Tolstonogov
研究了可分离Hilbert空间中具有可积非凸的整泛函在混合非凸约束下由非线性演化方程描述的控制系统解所定义的控制下的最小化问题。该系统的演化算子是一个固有的、凸的、下半连续函数随时间的次微分。在初始问题的基础上，考虑了控制约束为凸的松弛问题和被积函数相对于控制约束为凸的松弛问题。在充分一般的假设下，证明了松弛问题有一个最优解，并且对于任何最优解，存在一个关于轨迹和泛函收敛到最优解的初始问题的最小化序列。详细讨论了一类带障碍物的可控抛物型变分不等式。
We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints. The evolutionary operator of the system is the subdifferential of a proper, convex, lower semicontinuous function depending on time.Along with the initial problem, the author considers the relaxed problem with the convexicated control constraint and the integrand convexicated with respect to the control.Under sufficiently general assumptions, it is proved that the relaxed problem has an optimal solution, and for any optimal solution, there exists a minimizing sequence of the initial problem converging to the optimal solution with respect to trajectories and the functional. An example of a controlled parabolic variational inequality with obstacle is considered in detail.