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81

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Introduction

## Publications

Publications (81)

We consider the problem of the dynamic reconstruction of an observed state trajectory \(x^{*}(\cdot)\)
of an affine deterministic dynamic system and a control that has generated this trajectory.
The reconstruction is based on current information about inaccurate discrete measurements of \(x^{*}(\cdot)\).
A correct statement of the problem on the co...

The paper is devoted to the problem of dynamic control reconstruction for controlled deterministic affine systems. The reconstruction has to be carried out in real time using known discrete inaccurate measurements of an observed trajectory of the system. This trajectory is generated by an unknown measurable control with values in a given compact se...

Dynamic reconstruction problems are studied for controlled systems linear relative to controls. It is assumed that information about inaccurate current measurements of real states of the systems comes at discrete instants. The control generating this motion has to be reconstructed in real time. A new method is suggested to solve this inverse proble...

Basing on discrete inaccurate measurements of states of a controlled system, the problem of dynamic reconstruction of control generating the states is considered. A new original algorithm for solving this problem in real time is proposed. The solution of this inverse problem is obtained with the help of extremals in auxiliary problems of calculus o...

A chemotherapy model for a malignant tumor is considered, and the optimal control (therapy) problem of minimizing the number of tumor cells at a fixed final instant is investigated. In this problem, the value function is calculated, which assigns the value (the optimal achievable result) to each initial state. An optimal feedback (optimal synthesis...

The a posteriori analysis of the realized motions (e.g. trajectories and controls) is an important part of the theory of optimal control, variational calculus and decision making. This paper is devoted to solving inverse problems of reconstruction of realized controls for dynamic control systems using known history of inaccurate measurements of a r...

We consider dynamic reconstruction (DR) problems for controlled dynamical systems linear in controls and nonlinear in state variables as inaccurate current information about real motions is known. A solution of this on-line inverse problem is obtained with the help of auxiliary problems of calculus of variations (CV) for integral discrepancy functi...

In this paper a construction of the value function is obtained in the problem of chemotherapy of a malignant tumor growing according to the Gompertz law, when a therapy function has two maxima. The aim of therapy is to minimize the number of tumor cells at the given final instance.

Landing process on the Moon is under consideration. It is assumed that information about real motions is known with errors. A new algorithm for solving online dynamic reconstruction problems for controls of the navigation system is created. Key elements of the constructions are solutions of corresponding hamiltonian (characteristic) systems in auxi...

We consider the problem of identifying the parameters of a dynamic system from a noisy history of measuring the phase trajectory. We propose a new approach to the solution of this problem based on the construction of an auxiliary optimal control problem such that its extremals approximate the measurement history with a given accuracy. Using the sol...

The Cauchy problem for a nonlinear noncoercive Hamilton – Jacobi equation with state constraints is under consideration. Such a problem originates in molecular biology. It describes the process of evolution in molecular genetics according to the Crow – Kimura model. A generalized solution of prescribed structure is constructed and justifed via calc...

Perturbed inverse reconstruction problems for controlled dynamic systems are under consideration. A sample history of the actual trajectory is known. This trajectory is generated by a control, which isn’t known. Moreover, the deviation of the samples from the actual trajectory satisfies the known estimate of the sample error. The inverse problem wi...

A boundary value problem with state constraints is under consideration for a nonlinear noncoercive Hamilton-Jacobi equation. The problem arises in molecular biology for the Crow – Kimura model of genetic evolution. A new notion of continuous generalized solution to the problem is suggested. Connections with viscosity and minimax generalized solutio...

The Cauchy problem for the Hamilton–Jacobi equation with state constraints is considered. A justification for a construction of a generalized solution with given structure is provided. The construction is based on the method of characteristics and on solutions of problems related to calculus of variations.

An identification problem is considered as inaccurate measurements of dynamics on a time interval are given. The model has the form of ordinary differential equations which are linear with respect to unknown parameters. A new approach is presented to solve the identification problem in the framework of the optimal control theory. A numerical algori...

For systems linear in control, we consider problems of recovering the dynamics and control from a posteriori statistics of trajectory sampling and known estimates for the sampling error. An optimal control problem of minimizing an integral regularized functional of dynamics and statistics residuals is introduced. Optimal synthesis is used to constr...

The paper is devoted to the substantiation of the negative discrepancy method for solving inverse problems of the dynamics of deterministic control systems that are nonlinear in state variables and linear in control. The problem statements include the known sampling history of trajectories measured inaccurately, with known error estimates. The inve...

We introduced notions of generalized solutions of PDEs of the first order (Hamilton-Jacobi-Bellman equation and a quasi-linear hyperbolic system of the first order). Connections between the solutions and the value functions to optimal control problems are studied. We described properties of the solutions with the help of Cauchy characteristics meth...

In this paper, notions of global generalized solutions of Cauchy problems for the Hamilton–Jacobi–Bellman equation and for a quasilinear equation (a conservation law) are introduced in terms of characteristics of the Hamilton–Jacobi equation. Theorems on the existence and uniqueness of generalized solutions are proved. Representative formulas for g...

Perturbed inverse problems are under consideration for dynamical systems linear relative controls. It is assumed that sampling history and sampling error estimate are known. Auxiliary optimal control problems are introduced to minimize a regularized integral discrepancy functional. The trajectories of the system are constructed with the help of Opt...

We introduced notions of generalized solutions of PDEs of the first order (Hamilton-Jacobi-Bellman equation and a quasi-linear hyperbolic system of the first order). Connections between the solutions and the value functions to optimal control problems are studied. We described properties of the solutions with the help of Cauchy characteristics meth...

Problems on reconstruction of real motions of dynamical models are considered when statistic data on measurements of motions are given, and borders of admissible errors of the measurements are known. The dynamical models are described by ordinary differential equations depending on controlled parameters.
A new approach is suggested to solve the inv...

A Cauchy problem is considered for a Hamilton-Jacobi equation that preserves the Bellman type in a spatially bounded strip. Sufficient conditions are obtained under which there exists a continuous generalized (minimax/viscosity) solution to this problem with a given structure in the strip. A construction of this solution is presented.

The existence of continuous positional strategies of ɛ-optimal feedback is proved for linear optimal control problems with a convex terminal cost. These continuous feedbacks are determined from Bellman's equation in ɛ-perturbed control problems with an integral-terminal cost and a smooth value function. An example is given in which an ɛ-optimal con...

We consider optimal control problems of prescribed duration. A new numerical method is suggested to solve the problems of
controlling until a given instant. The solution is based on a generalization of the method of characteristics for Hamilton–Jacobi–Bellman
equations. Constructions of optimal grid synthesis are suggested and numerical algorithms...

The value function (or the optimal result function) arising in optimal control problems with the Bolza pay-off functionals is studied as the unique minimax or viscosity solution of a corresponding boundary problem for the Hamilton-Jacobi-Bellman (the dynamic programming equiation) equation. It is obtained for one-dimensional state space, that the c...

Optimal control problems with a terminal pay-off functional are considered. The dynamics of the control system consists of rapid oscillatory and slow non-linear motions. A numerical method for solving these problems using the characteristics of the Hamilton–Jacobi–Bellman equation is presented. Estimates of the accuracy of the method are obtained....

We consider optimal control problems with fixed final time and terminal-integral cost functional, and address the question
of constructing a grid optimal synthesis (a universal feedback) on the basis of classical characteristics of the Bellman equation.
To construct an optimal synthesis, we propose a numerical algorithm that relies on the necessary...

In the paper, we consider nonlinear optimal control problems with the Bolza functional and with fixed terminal time. We suggest
a construction of optimal grid synthesis. For each initial state of the control system, we obtain an estimate for the difference
between the optimal result and the value of the functional on the trajectory generated by the...

Presented was a new grid method for design the feedback-optimal control law (optimal design) in the problem of optimal control
of prescribed duration. The difference between the optimal result and the result of control based on the proposed grid design
was estimated for any initial state of the controlled system. These estimates were illustrated by...

n>A generalization of the method of characteristics for Bellman equation is applied to solving optimal control problems with terminal cost functional. A new numerical method is suggested for constructing optimal open-loop controls. The method is based on the backward procedure of integrating the characteristic system ODE's. This method is tested in...

Necessary and sufficient conditions for a minimax solution to the Cauchy problem for the Hamilton-Jacobi-Bellman equation
are obtained as viability conditions for classical characteristics inside the graph of this solution. Using this property,
a representative formula for a one-dimensional conservation law in terms of classical characteristics is...

We propose a numerical algorithm for constructing an optimal synthesis in the control problem for a nonlinear system on a
fixed time interval. We estimate the difference between the values of the cost functional on optimal trajectories and on the
trajectories constructed according to this algorithm. The operation of the algorithm is illustrated by...

Optimal control problems under conflict or uncertainties are considered on a finite time interval. Resolving dynamical procedures are suggested. They are based either on constructions of stable bridges produced with the help of step-by-step programmed absorptions or on applications of the backward dynamic programming method. Results of simulations...

Optimal control problems under conflict or uncertainties are considered on a finite time interval. Resolving dynamical procedures are suggested. They are based either on constructions of stable bridges produced with the help of step-by-step programmed absorptions or on applications of the backward dynamic programming method. Results of simulations...

In this paper, a solution algorithm for the optimal control problem for the system with uncertain dynamics which is the asymptotics
of the singularly perturbed system with fast oscillator is proposed.

The Dirichlet problems for singularly perturbed Hamilton–Jacobi–Bellman equations are considered. Some impulse variables in the Hamiltonians have coefficients with a small parameter of singularity ε in denominators.The research appeals to the theory of minimax solutions to HJEs. Namely, for any ε>0, it is known that the unique lower semi-continuous...

A new numerical method is suggested for constructing the value function in optimal control problems of prescribed duration
with positional running cost along motions of controlled dynamical systems. The algorithm is based on a backward procedure
involving characteristics of the Bellman equation. Estimations of the approximation are provided. Result...

In this paper we consider closed-loop control problems. We formulate necessary and sufficient conditions which must be satisfied by the value function (the optimal result function). The conditions proposed here are formed of two differential inequalities and the boundary conditions. These inequalities are infinitesimal forms of the so-called stabil...

Singularly perturbed differential games with "fast" and "slow" motions and the Bolza type payoff functionals are considered. Sufficient conditions are obtained for the value functions of the games to converge to the value function of the asymptotic unperturbed game as a parameter of singularity £ tends to O.

Optimal control problems with nonsmooth value functions are condidered. It is obtained that universal optimal feedbacks to the problems can be constructed in two ways. The first constructon is based on pointwise limits of ε-optimal feedbacks (suboptimal controls). The second one has involved limits of arguments of minimum operation in the Bellman e...

Unlike the previous investigation of the sufficient conditions for the convergence of minimax solutions of singularly perturbed Hamilton-Jacobi (H-J) equations, a typical example of which would be the Bellman-Isaacs (B-I) equations, convergence conditions are formulated not in terms of auxiliary constructs [1], but in terms of the Hamiltonian, the...

Sufficient conditions under which the solutions of the Cauchy problem for singularly-perturbed Hamilton-Jacobi equations will converge to a limit are established. The results are used to investigate the asymptotic behaviour of the value function of a differential game involving fast and slow motions.

Some new results concerning a generalized solution theory for the partial differential equations of the first order are reviewed shortly. Main attention is paid to the studies of the minimax solutions. The main conceptions used in these investigations are presented. The connection of the minimax solutions with constructions of the differential posi...

A concise survey of some new results in the theory of generalized solutions of first-order partial differential equations is presented. Basic attention is paid to the investigation of minimax solutions. The basic concepts used in these investigations are presented, the connection between minimax solutions and the constructs of positional differenti...

An important connection between the Pontryagin maximum principle and the Bellman dynamic programming is established. It is proved that adjoint variables appearing in the maximum principle conditions are generalized gradients of the value function, which is the “viscosity” solution of the Hamilton-Jacobi-Bellman equation. This relationship completes...

A position differential game with fixed termination instant is examined. Stability properties are investigated, consisting mainly in the formulation of necessary and sufficient conditions satisfied by the differential game's value function (potential). The infinitesimal form of the stability properties leads to differential inequalities generalizin...

We examine a differential game in which the players can control the system's motion with the aid of generalized impulses. Similar problems were investigated in [1–5], In this paper we describe position procedures of control with a guide, within the framework of which we establish alternative conditions for the solvability of encounter and evasion p...

Optimal control problems with Bolza type functionals are considered as positional differential games with imaginary second player. The value function of the game is nonsmooth. It is the unique minimax (and/or viscosity) solution to the Isaacs-Bellman equation. Sufficient conditions for optimal feedbacks are obtained on the base of the corresponding...

The value function (or the optimal result function) arising in optimal control problems with the Bolza pay-off functionals is studied as the unique minimax or viscosity solution of a corresponding boundary problem for the Hamilton-Jacobi-Bellman (the dynamic programming equiation) equation. It is obtained for one-dimensional state space, that the c...

An approximation problem is considered for minimax solutions of unperturbed Hamilton-Jacobi equations (HJE) via minimax solutions of singularly perturbed HJEs in an augmented phase space. The singularly perturbed Hamiltonians have a small parameter in denominators of terms containing additional impulse variables. Sufficient conditions for the appro...

## Projects

Project (1)

The proposed project involves the development of a theory and numerical methods for solving boundary value problems for the Hamilton-Jacobi-Bellman equations in nonclassical statements (the absence of the coercivity condition for the Hamiltonian, the Hamiltonian discontinuity in the phase variable, the hybridity of the Hamiltonian, nonstandard boundary conditions, and the domain of solution). It is also proposed to study the connection between the generalized solutions of these equations and the solutions of the dual problems of the calculus of variations and dynamic optimization.