Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

Error bounds for attainability sets of control systems with phase constraints

The problem of the error bounds for attainability sets of control systems described by ordinary differential equations under discretization of the phase constraints is studied. The peculiarity of the problems investigated in this paper is the phase constraints in the form of equality.     

Non-singular locally optimal ellipsoidal approximation of the estimate of the states of linear systems

A problem which arises when estimating the attainability domains of linear dynamical systems by ellipsoids is investigated in a short time interval in the case when the initial position of the system in phase space is known precisely for some at least coordinates. A method is proposed which allows one to avoid problems associated with the degeneracy of the right-hand sides of the differential equations of the locally optimal ellipsoidal approximation. The mathematical meaning of these equations is made more precise in the case of the minimization of the phase volume. An example is given.     

Internal ellipsoidal estimates of attainability sets for linear systems under conditions of uncertainty

The problem of constructing internal ellipsoidal estimates of the geometric difference between two ellipsoids and applying the estimated results for the attainability sets of linear systems with a disturbance is considered. An addition to the existing method of constructing the difference between two ellipsoids is presented, and the previous constraints are removed. In the process of validating the addition, some relationships between certain properties of constructed ellipsoidal estimations and set convexity are given, being the data for the problem. A method for estimating the attainability sets for linear systems with a disturbance, equivalent to the existing approach to systems without disturbances, are given. The disturbances are considered using the obtained results.     

The optimal choice of control constraints

The problem of the optimal choice of the limits of a set of possible values of the control during motion for the purpose of obtaining the required form of the attainability set of a linear dynamical system in a specified time interval is considered. Using the method, in which these sets are approximated by ellipsoids, the problem of controlling the parameters of the ellipsoid containing the control vector is solved. Then a functional, which depends on the matrix of the ellipsoid, containing the phase vector, reaches its maximum. The order in which the corresponding formulae are used is illustrated using the example of a simple mechanical system. The results obtained are suitable for systems in which, instead of the control vector, there is an interference vector with controllable boundaries of possible changes and can be extended to stochastic systems.     

Solving the controllability problem for a nonlinear three-dimensional system

The controllability problem is solved for a nonlinear system of three differential equations that simulates aerated biological wastewater treatment using the thermal metabolism of thermophilic aerobic bacteria. Properties of the phase variables of such a system, bounded and continuable to a given segment, are established. The corresponding attainability set is parameterized by the switching instants of piecewise-constant controls. Parameterization helps reduce the controllability problem for the relevant system to finite dimensional conditional minimization of the auxiliary function. The results from numerical calculations for the solution to the initial problem are given.     

Estimation of the attainability set for a linear system based on a linear matrix inequality

The problem under consideration is the construction of an attainability set’s internal approximation for a full controllable linear time-invariant system. This approximation is obtained as an intersection of two domains given by quadratic forms. One of these forms is based on parameters of the original system. The other form is produced by the solution to some linear matrix inequality. The method proposed here is illustrated by a numerical example.     

An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates are obtained.     

Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.     

有交易费的折算资产优化可达性

针对所给出的有交易费的资产模型，引入了资产折算函数，并利用辅助鞅和凸分析方法，讨论了该模型下一种资产优化的可达性。     

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.     

Representation of attraction elements in abstract attainability problems with asymptotic constraints

We consider an abstract attainability problem with asymptotic constraints in a topological space. We construct an extension in the class of ultrafilters of widely interpreted measurable spaces. We study an example of a static problem on the asymptotic attainability in the class of layer functions.     

On convex closed bounded bodies without farthest points such that the closure of their complement is antiproximinal

We consider an abstract attainability problem with constraints of asymptotic nature defined in the form of a nonempty family of subsets in the space of ordinary solutions. Various variants of implementing asymptotic effects are considered (convergence in a topological space, cycles, and so on). A rather general method is suggested for presenting the results of action of approximate solutions; this method generalizes constructions based on sequences in the space of ordinary solutions.     

Complexity of approximation of Lipschitz functions

The attainability of the lower bound by order is proved for the complexity of a problem of approximation of Lipschitz functions by diagrams of functional elements in bases consisting of a finite number of Lipschitz functions and a continuum of constants from a bounded set.     

Estimation of statistical characteristics of attainability sets of controllable systems

We study an expansion of the notion of invariance for sets with respect to controllable systems and differential inclusions. Namely, we study statistically invariant sets and statistical characteristics of attainability sets of controllable systems. We obtain a lower bound for the lower relative frequency of the absorption of the attainability set of a system by a given set and establish new sufficient conditions of the statistical invariance of the set with respect to the controllable system. We give examples of the calculation of statistical characteristics for the linear Cauchy problem and a linear controllable system with almost periodic coefficients.     

A method of optimal real-time computation of a linear system with retarded control

A new method of solving time-optimal control problems in real time is developed. The method is based on the following: 1) approximating the attainability sets with a family of hyperplanes; 2) subdividing the whole computational process into the computations performed beforehand and those that are carried out while the control takes place; 3) integrating the differential equations only over the displacement intervals of the final time point and the switching time points. The computational cost of the method is evaluated. The peculiarities of calculating the optimal control of a linear system with retarded control in real time are considered. The results of simulation and numerical calculations are presented.     

On the Optimal Correction of Infeasible Systems of Linear Inequalities

We study the optimum correction of infeasible systems of linear inequalities through making minimal changes in the coefficient matrix and the right-hand side vector by using the Frobenius norm. It leads to a special structured unconstrained nonlinear and nonconvex problem, which can be reformulated as a one-dimensional parametric minimization problem such that each objective function corresponds to a trust region subproblem. We show that, under some assumptions, the parametric function is differentiable and strictly unimodal. We present optimally conditions, propose lower and upper bounds on the optimal value and discuss attainability of the optimal value. To solve the original problem, we propose a binary search method accompanied by a type of Newton–Lagrange method for solving the subproblem. The numerical results illustrate the effectiveness of the suggested method.

     

CONSTRUCTION OF APPROXIMATE ATTAINABILITY SETS FOR LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL INCLUSIONS

We present a numerical method for constructing, with a specified accuracy attainability sets for Lipschitzian quantum stochastic differential inclusions. Results here generalize the Komarov-Pevchikh results concerning classical differential inclusions to the present noncommutative quantum setting involving unbounded linear operators on a Hilbert space.

AMS Subject Classification (1991): 60H10, 60H20, 65L05, 81S25.