L1\mathcal{L}^1 limit solutions for control systems

For a control Cauchy problem $$\dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha,\quad x(a)=\bar x,$$ on an interval $[a,b]$, we propose a notion of limit solution $x,$ verifying the following properties: i) $x$ is defined for $\mathcal{L}^1$ (impulsive) inputs $u$ and for standard, bounded measurable, controls $v$; ii) in the commutative case (i.e. when $[g_{\alpha},g_{\beta}]\equiv 0,$ for all $\alpha,\beta=1,...,m$), $x$ coincides with the solution one can obtain via the change of coordinates that makes the $g_\alpha$ simultaneously constant; iii) $x$ subsumes former concepts of solution valid for the generic, noncommutative case. In particular, when $u$ has bounded variation, we investigate the relation between limit solutions and (single-valued) graph completion solutions. Furthermore, we prove consistency with the classical Carath\'eodory solution when $u$ and $x$ are absolutely continuous. Even though some specific problems are better addressed by means of special representations of the solutions, we believe that various theoretical issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and endpoint constraints, for which no extra assumptions (like e.g. coercivity, bounded variation, commutativity) are made in advance

Variational Optimality Conditions with Feedback Descent Controls that Strengthen the Maximum Principle

We derive nonlocal necessary optimality conditions that strengthen both classical and nonsmooth Maximum Principles for nonlinear optimal control problems with free right-hand end of trajectories. The strengthening is due to employment of feedback controls, which are assumed to ensure a descent of a value of the cost functional, and are extremal with respect to certain solutions of a Hamilton – Jacobi inequality for weakly monotone functions. The main results are Feedback Minimum Principles for smooth and nonsmooth problems, that are formulated through accessory dynamic optimization problems. Effectiveness of these necessary optimality conditions are illustrated by examples

Проектування оптимальної якорної системи бурового судна

The drillship multianchor system design for is reduced to nonstandard optimization problem concerning the extremum seeking of the function of several variables; the algorithm of the problem solving is given and its numerical illustration is offered for two values of water depth: 25 m and 200 m.Задача проектирования многоякорной системы бурового судна сведена к нестандартной оптимизационной задаче о поиске экстремума функции нескольких переменных в ограниченной области ее определения; приведены описание алгоритма решения задачи и численный пример его реализации для двух значений глубины водоема – 25 и 200 м.Задача проектування багатоякірної системи бурового судна зведена до нестандартної оптимізаційної задачі про пошук екстремуму функції декількох змінних в обмеженій області її визначення; наведені опис алгоритму розв'язання задачі і числовий приклад його реалізації для двох значень глибини водоймища – 25 і 200 м

Feedback Minimum Principle for Quasi-Optimal Processes of Terminally-Constrained Control Problems

In a series of previous works, we derived nonlocal necessary optimality conditions for free-endpoint problems. These conditions strengthen the Maximum Principle and are unified under the name “feedback minimum principle”. The present paper is aimed at extending these optimality conditions to terminally constrained problems. We propose a scheme of the proof of this generalization based on a “lift” of the constraints by means of a modified Lagrange function with a quadratic penalty. Implementation of this scheme employs necessary optimality conditions for quasi-optimal processes in approximating optimal control problems. In view of this, in the first part of the work, the feedback minimum principle is extended to quasi-optimal free-endpoint processes (i.e. strengthen the so-called $\varepsilon$-perturbed Maximum Principle). In the second part, this result is used to derive the approximate feedback minimum principle for a smooth terminally constrained problem. In an extended interpretation, the final assertion looks rather natural: If the constraints of the original problem are lifted by a sequence of relaxed approximating problems with the property of global convergence, then the global minimum at a feasible point of the original problem is admitted if and only if, for all $\varepsilon>0$, this point is $\varepsilon$-optimal, for all approximating problems of a sufficiently large index. In respect of optimal control with terminal constraints, the feedback $\varepsilon$-principle serves exactly for realization of the formulated assertion