An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.     

Quasilinear Pursuit Differential Game with a Simple Dynamics under Phase Constraints

On a fixed closed time interval we consider a quasilinear pursuit differential game with a convex compact target set under a phase constraint in the form of a convex closed set. We construct a convex compact guaranteed capture set similar to an alternating Pontryagin sum and define the guaranteed piecewise-programmed strategy of the pursuer ensuring the hitting of the target set by the phase vector satisfying the phase constraint in finite time. Under certain conditions, we prove the convergence of the constructed alternating sum in the Hausdorff metric to a convex compact set, which is an analog of the alternating Pontryagin integral for the differential game.     

A dynamic programming method for controlled linear systems with delays

A linear system with permanent delay is considered. A method of dynamic programming for constructing attainability sets and solving the problem of target control for the systems is used. The expressions for value functionals described by solutions to the corresponding Hamilton-Jacobi-Bellman equation are obtained. It is proved that these value functionals calculated by means of convex analysis satisfy the above equations. Strategies for synthesized control for the problem of hitting on the target set are given.     

A control problem under incomplete information for a linear stochastic differential equation

A problem of guaranteed closed-loop control under incomplete information is considered for a linear stochastic differential equation (SDE) from the viewpoint of the method of open-loop control packages worked out earlier for the guidance of a linear control system of ordinary differential equations (ODEs) to a convex target set. The problem consists in designing a deterministic open-loop control providing (irrespective of a realized initial state from a given finite set) prescribed properties of the solution (being a random process) at a terminal point in time. It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problem for the SDE is reduced to an equivalent problem for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. Solvability conditions for the problems in question are written.     

Time optimal control computation with application to ship steering

An efficient computational scheme for solving a general class of linear time optimal control problems, where the target set is a compact and convex set with nonempty interior in the state space, is presented. The scheme is applied to solve the ship steering control problem, and excellent results are obtained.     

The infima of functionals of a special kind on a compact convex set

It is proved that the infimum of the ratio of a concave functional and a convex functional is attained at the extreme points of a compact convex set in a normed linear space. A criterion for the membership of a given element in the set of extreme points is proposed and the existence of a strongly convex functional on a compact set is shown.     

Guidance problem for a distributed system with incomplete information on the state coordinates and an unknown initial state

We study the problem of guaranteed positional guidance of a linear partially observable control system with distributed parameters to a convex target set at a given time. The problem is considered under incomplete information. More precisely, we assume that the system is subjected to an unknown disturbance; in addition, the initial state is assumed to be unknown as well. Further, the sets of admissible disturbances and the set of admissible initial states, which is assumed to be finite, are known. An algorithm for solving the problem is suggested.     

平面有限点集中的最大面积六边形

若平面上的有限点集构成凸多边形的顶点集,则称此有限点集处于凸位置令P表示平面上处于凸位置的有限点集,研究了P的子集所确定的凸六边形的面积与CH(P)面积比值的最大值问题.     

具非线性控制变量的非自治松驰系统时间最优追踪控制问题中的若干结果

本文讨论了一类广义非自治离散松驰系统的时间最优控制问题，将R^n中点曲线的目标约束推广为凸集值函数的超曲线约束．在证明了松驰系统与原系统可达集相等的基础上，得到了最优控制的存在性．由凸集分离定理及终端时间闺值函数方程，我们获得了最大值原理及最优控制时间的确定方法．较之Hamilton方法，本文的条件更一般．离散松驰系统的相关结论可以用于分散控制．     

Iterative algorithm for minimizing a convex function at the intersection of a spherical surface and a convex compact set

A numerical algorithm for minimizing a convex function on the set-theoretic intersection of a spherical surface and a convex compact set is proposed. The idea behind the algorithm is to reduce the original minimization problem to a sequence of convex programming problems. Necessary extremum conditions are examined, and the convergence of the algorithm is analyzed.     

A criterion for checking if a convex set belongs to the union of a finite number of convex sets

A necessary and sufficient condition for the inclusion of a convex compact set in the union of a finite number of convex sets is proved. This condition obtained using the convex analysis techniques is a condition on the maximin of a given function. Using the dynamic programming, checking this condition is reduced to evaluating a set of functions and checking a condition for their values. The reduced form of the criterion is more convenient from the computational point of view.     

Investigating One Class of Extended Open-Loop Guidance Problems

An alternative solvability criterion and guiding control construction procedure are proposed in one class of extended open-loop guidance problems. A control system is described by linear differential equations. The control domain is a rectangular parallelepiped, and the convex and closed target set has a special form. An equivalent system of linear inequalities is constructed for the extended open-loop guidance problem, and a correspondence between their solutions is established. The difference between the developed scheme and the known research techniques of this class of problems is the possibility of solvability criterion verification and search for guiding control in analytical form in a finite number of steps. The proposed technique for solving the extended openloop guidance problem is illustrated using a concrete example.     

Lower bounds and duality in the optimal control of distributed systems

We develop a duality theory for problems of minimization of a convex functional on a convex set and apply this theory to optimal control problems with non-differentiable functional and state as well as control constraints. The dual problem is sometimes found to be simpler than the primal problem. The equivalence of the primal and the dual problem is established for problems in which the functional is strongly convex. A posteriori error are also given for such problems.     

LaGO: a (heuristic) Branch and Cut algorithm for nonconvex MINLPs

We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot be applied, and we are restricted to sampling techniques in case of nonquadratic nonconvex functions. The linear relaxation is further improved by mixed-integer-rounding cuts. Also box reduction techniques are applied to improve efficiency. Numerical results on medium size test problems are presented to show the efficiency of the method.     

The optimization of a guaranteed result with a delay in the control

The problem of feedback control with a quality index in the form of the Euclidean norm of a set of deviations of the trajectory of motion of the system from the origin of coordinates at specified instants of time is considered for a linear dynamical system that is subjected to the effects of noise and contains a delay in the control element. A procedure for calculating the value of the optimal guaranteed result based on the recursive construction of upper convex hulls of auxiliary functions is given. A method for shaping the controlling actions that guarantee the achievement of this result is described.     

A level set algorithm for a class of reverse convex programs

A new algorithm is presented for minimizing a linear function subject to a set of linear inequalities and one additional reverse convex constraint. The algorithm utilizes a conical partition of the convex polytope in conjuction with its facets in order to remain on the level surface of the reverse convex constraint. The algorithm does not need to solve linear programs on a set of cones which converges to a line segment.     

Numerical Algorithm for Minimizing a Convex Function on the Intersection of a Smooth Surface and a Convex Compact Set

Computational Mathematics and Mathematical Physics - A numerical algorithm for minimizing a convex function on the set-theoretic intersection of a smooth surface and a convex compact set in...     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.     

Optimal Synthesis in a Model Problem with Two-Dimensional Control Lying in an Arbitrary Convex Set

Mathematical Notes - We consider a model nilpotent convex problem with two-dimensional control from an arbitrary convex set Ω. For the case in which Ω is a polygon, the problem is solved...