We study the optimal impulse control problem with quadratic perfomance functional for a descriptor system. The system evolution is described by a linear differential-algebraic equation not solved with respect to the derivative of the state. The system is controlled by changing the measurable control and the pure impulse control. The pure impulse control is characterized by impulse intensities and moments of impulse applications. The main restriction is that the characteristic matrix pencil corresponding to the state equation is regular. In terms of the characteristic matrix pencil, we establish the conditions for the existence and uniqueness of the optimal control and the corresponding optimal state. The optimal control and the optimal state are constructed by using the adjoint state, which is a solution to the adjoint two-point boundary value problem. The results are illustrated on an example of descriptor system that describes transient states in a radio technical filter. For this system, we consider an energetic perfomance functional with impulse intensities characterizing the energy of inertial elements and input voltage of the filter and also moments of impulse applications. Transient states under impulsive perturbations of currents and voltages are described by using the formula of variation of constants for the impulsive descriptor system.

A differential pursuit game in a stochastic descriptor linear system is analyzed. The dynamic of the system is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the stochastic formula of the variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct pursuer’s control bringing the dynamic vector of the system to the terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transients in a radio engineering filter with random disturbances in the form of white noise.

Results on properties of a class of descriptor control systems are obtained. The system state is described by a nonlinear impulsive delay equation not solved with respect to the derivative of the state. The matrix coefficient in the time derivative is allowed to be noninvertible. We indicate conditions to solve this equation. The conditions are formulated in terms of the resolvent of characteristic matrix pencil. The results are illustrated on an example of a descriptor system that describes transient states in a radio technical filter with lossless transmissions.

We study a differential pursuit game in a system dynamically described by a linear functional differential equation. The coefficients of the equation are closed linear operators acting in Hilbert spaces. The operator at the derivative of state depends on the current time and is, generally speaking, not invertible. Our main assumption is a restriction imposed on the characteristic operator pencil of the equation on a ray of the real positive semiaxis. The solutions of the equation are represented with the help of the formula of variation of constants in which the effect of delay is taken into account as a result of summation of shift-type operators. To establish conditions under which the dynamic vector of the system approaches a cylindrical terminal set, we use constraints imposed on the support functionals of two sets determined by the behaviors of the pursuer and evader. We also present an example of differential game in a pseudoparabolic system described by a partial functional-differential equation.

УДК 517.9Вивчається диференцiальна гра переслiдування у системi, динамiка якої описується лiнiйним функцiонально-диференцiальним рiвнянням. Коефiцiєнти рiвняння є замкненими лiнiйними операторами, що дiють у гiльбертових просторах. Оператор при похiднiй стану у поточний час є, взагалi кажучи, необоротним. Основне припущення полягає в обмеженнi на характеристичну операторну в’язку рiвняння на променi дiйсної додатної пiвосi. Розв’язки рiвняння зображуються за допомогою формули варiацiї сталих, де ефект запiзнення враховується шляхом пiдсумовування операторiв типу зсуву. Для отримання умов наближення динамiчного вектора системи до цилiндричної термiнальної множини ми використовуємо обмеження на опорнi функцiонали двох множин, що визначаються поведiнками переслiдувача i втiкача. Наведено приклад диференцiальної гри в псевдопараболiчнiй системi, що описується функцiонально-диференцiальним рiвнянням з частинними похiдними.

We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

A differential pursuit game in a descriptor system is analyzed. The evolution of the system is described by an algebraic linear differential equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. The technique of set-valued mappings and their selectors, as well as constraints on the functionals defined by the behaviors of the pursuer and evader are used. The paper contains examples to illustrate a differential game in radio engineering systems. In particular, conflict-controlled transients in four-pole filters are analyzed.

We study the game problem of approach for a system whose dynamics is described by a stochastic differential equation in a Hilbert space. The main assumption on the equation is that the operator multiplying the system state generates a strongly continuous semigroup (a semigroup of class C-0). Solutions of the equation are represented by a stochastic variation of constants formula. Using constraints on the support functionals of sets defined by the behavior of the pursuer and the evader, we obtain conditions for the approach of the system state to a cylindrical terminal set. The results are illustrated with a model example of a simple motion in a Hilbert space with random perturbations. Applications to distributed systems described by stochastic partial differential equations are considered. By taking into account a random external influence, we consider the heat propagation process with controlled distributed heat sources and sinks.

We study the optimal control problem for a descriptor system whose evolution is described by Ito’s differential-algebraic equation. The quadratic cost functional is considered. The main constraint is that the characteristic matrix pencil corresponding to the equation is regular. We establish the conditions for the existence and uniqueness of the optimal control and the corresponding optimal state. The results are illustrated on an example of a descriptor system that describes transient states in a radio engineering filter with random perturbations in the form of white noise.

We establish the conditions to decompose a complex descriptor control system into simpler subsystems. The system state and input are described by equations not solved with respect to the derivative of the state. We consider two types of decompositions: sequential and parallel ones. The decomposition conditions are formulated in terms of the existence of invariant pairs of subspaces for operator pencils consisting of system coefficients. The results are illustrated by the example of a descriptor system that describes transient states in a radio-engineering filter. We perform the cascade–parallel decomposition of forth-order filter into the simplest first-order filters, each containing one inertial element.

We study a differential game of approach in a system whose dynamics is described by a Sobolev‐type second‐order operator differential equation in Hilbert spaces. Solutions of the equation are represented by cosine and sine operator functions. To obtain solvability conditions of the game problem, we use support functionals of two sets, which defined by the behaviors of pursuer and evader. The results are applied to the investigation of conflicted‐controlled dynamics of bending waves in a rod.

To analyze complex linear descriptor control systems described by relations unsolved with respect to the state derivative there is performed their decompositions into chains of simpler systems. In terms of invariant pairs of subspaces of characteristic and perturbed operator pencils of the system we establish the conditions to represent the system as a series connection of systems of smaller dimensions. The results are illustrated by examples of descriptor systems that describe transient modes in radiotechnical filters.

The optimal control problem for a system whose evolution is described by a Sobolev-type second-order retarded operator-differential equation is studied. The main assumption is that a restriction is imposed on the derivatives of the resolvent of the quadratic operator pencil on a ray in the right half-plane. Several applications to systems described by non-Kovalevskaya-type partial differential equations are considered.

We consider a game problem of approach for a system whose dynamics is described by a partial differential equation not of Kovalevskaya type, i.e., unsolved with respect to the time derivative. The equation with boundary conditions is written in a Hilbert function space in an abstract form as a differential operator equation. Using the method of resolving functionals, we obtain sufficient conditions for the approach of the system’s dynamical vector to a cylindrical terminal set. The results are exemplified by means of a model problem concerning a filtering process for a fluid in fractured porous rocks.

We consider the game problem of approach for a system whose dynamics is described by a differential operator equation in a Hilbert space. The equation is written in an implicit form with generally noninvertible operator multiplying the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in a right half-plane. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of the system to a cylindrical terminal set. Applications to systems described by partial differential equations are considered.

We study a differential game of approach in a system whose dynamics is described by an implicit differential-operator equation unsolved for the time derivative. The coefficients of the equation are closed operators on Hilbert spaces. We consider applications to systems with distributed parameters described by partial differential equations that are not of the Kovalevskaya type.

We establish conditions for the existence and uniqueness of the solutions of nonlinear functional-differential equations with impulsive action in a Banach space. The equation under consideration is not solved for the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in the right half-plane. Applications to partial functional-differential equations not of Kovalevskaya type are considered.

We study the game approach problem for a distributed system which is described by a linear differential-operator equation of Sobolev type. To study the differential game the method of resolving functions is extended to the case of Hilbert spaces. The application to partial differential equations of not Kovalevskaya type is considered, in particular, to the investigation of filtering fluids in fractured-porous rocks.