The Stabilization of the Solution of an inverse Problem for the Pseudoparabolic Equation

The asymptotic behavior of the strong solution to the inverse problem on recovering an unknown coefficient k(t) in a pseudoparabolic equation (u + ηMu) t + Mu + k(t)u = f is investigated. The differential operator M of the second order with respect spacial variables is supposed to be elliptic and selfajoint. It is proved that the solution of the inverse problem stabilizes to the solution of the appropriate stationary inverse problem as t → + ∞.

On boundary optimal control of a coefficient in a nonlinear parabolic equation

Работа связана с изучением нелинейных параболических систем, возникающих при моделировании и управлении физико-химическими процессами, в которых происходят изменения внутренних свойств материалов. Исследовано оптимальное управление одной из таких систем, которая включает в себя краевую задачу третьего рода для квазилинейного параболического уравнения с неизвестным коэффициентом при производной по времени, а также уравнение изменения по времени этого коэффициента. Обоснована постановка оптимальной задачи с финальным наблюдением искомого коэффициента, в которой управлением является граничный режим на одной из границ области. Получено явное представление дифференциала минимизируемого функционала через решение сопряженной задачи. Доказаны условия ее однозначной разрешимости в классе гладких функций. Полученные результаты имеют практическое значение для приложений в различных технических областях, медицине, геологии и т.п. Приведены некоторые примеры таких приложений. The work is connected with investigation of nonlinear parabolic systems arising in the mathematical modeling and control of physical-chemical processes in which inner properties of materials are subjected to changes. We consider optimal control in one of such systems that involves a boundary value problem of the third kind for a quasilinear parabolic equation with an unknown coefficient at the time derivative and, moreover, an additional equation for a time dependence of this coefficient. The optimal problem with a boundary control regime is justified for the given final observation of the sought coefficient. The exact representation for the differential of the minimization functional in terms of the solutions of the conjugate problem is obtained. The form of this conjugate problem and conditions of unique solvability in a class of smooth functions are shown. The obtained results are important for applications in various technical fields, medicine, geology, etc. Some examples of such applications are discussed.

On an Inverse Problem for the Stationary Equation with a Boundary Condition of the Third Type

The identification of an unknown coefficient in the lower term of elliptic second-order differential equation Mu + ku = f with the boundary condition of the third type is considered. The identification of the coefficient is based on integral boundary data. The local existence and uniqueness of the strong solution for the inverse problem is proved

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.

Direct tuning of gain-scheduled controller for electro-pneumatic clutch position control

This study proposes a gain-scheduled controller with direct tuning for the position control of a pneumatic clutch actuator that is installed in heavy-duty trucks. Pneumatic clutch actuators are highly nonlinear systems and cannot be easily controlled. Industries require a simple controller design that is easy to understand and requires few trial-and-error calibrations. Therefore, we adopted a gain-scheduled proportional integral derivative (PID) control law, which is a well-known and easy-to-understand nonlinear control method. In this approach, a gain scheduler is expressed using polynomials composed of coefficient parameters and controlled object states. The unknown coefficient parameters of the polynomials are directly tuned from the controlled object input/output data without having to use a controlled object model. The proposed controller design procedure is simple and does not require system identification or trial-and-error tuning. The effectiveness of the proposed method is verified by an experiment using an actual vehicle. The experimental results confirm the effectiveness of the proposed method for the position control of pneumatic clutch actuators.

Inverse Problem for the Sobolev Type Equation of Higher Order

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study.

On the definition of a time-dependent lower coefficient in a third-order hyperbolic equation

В работе рассматривается обратная задача для гиперболического уравнения третьего порядка. Ставится обратная задача, состоящая в определении неизвестного коэффициента, зависящего от времени. В качестве дополнительной информации для решения обратной задачи задаются значения решения задачи во внутренней точке. Доказывается теорема существования и единственности решения обратной задачи. Доказательство основано на выводе нелинейной системы интегральных уравнений типа Вольтерра второго рода и доказательстве его разрешимости. The paper deals with an inverse problem for a hyperbolic equation of the third order. An inverse problem is posed, which consists in determining an unknown coefficient that depends on time. As additional information for solving the inverse problem, we set the values of the solution to the problem at an interior point, and prove the existence and uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation of a nonlinear system of integral equations of the Volterra type of the second kind and the proof of its solvability.

Estimation for Varying Coefficient Models with Hierarchical Structure

The varying coefficient (VC) model is a generalization of ordinary linear model, which can not only retain strong interpretability but also has the flexibility of the nonparametric model. In this paper, we investigate a VC model with hierarchical structure. A unified variable selection method for VC model is proposed, which can simultaneously select the nonzero effects and estimate the unknown coefficient functions. Meanwhile, the selected model enforces the hierarchical structure, that is, interaction terms can be selected into the model only if the corresponding main effects are in the model. The kernel method is employed to estimate the varying coefficient functions, and a combined overlapped group Lasso regularization is introduced to implement variable selection to keep the hierarchical structure. It is proved that the proposed penalty estimators have oracle properties, that is, the coefficients are estimated as well as if the true model were known in advance. Simulation studies and a real data analysis are carried out to examine the performance of the proposed method in finite sample case.

Inverse coefficient problem for the time-fractional diffusion equation

We study the inverse problem of determining the time depending reaction diffu- sion coefficient in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is used and properties of this solution are investigated. The fundamental solution contains the Fox’s H− functions widely used in fractional calculus. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient which will be used in study inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.