Sufficient Conditions For Solvability Research Articles

Necessary and sufficient conditions for the spectra of the Sturm–Liouville operators with frozen argument

In this paper, we consider the inverse problem of recovering the Sturm–Liouville operator with frozen argument from its spectrum. In the case of irrational frozen argument, necessary and sufficient conditions for solvability of the inverse problem remained an open question. Here, we give a complete answer to this question. For this purpose, we suggest a new unified approach to studying inverse problems for operators with frozen argument, which is effective in both irrational and rational cases. Applying it, we obtain new-type asymptotic formulae completely characterizing the class of sequences that can be the spectra of such operators.

Passivity Analysis of Fractional-Order Neutral-Type Fuzzy Cellular BAM Neural Networks with Time-Varying Delays

In this paper, passivity analysis of fractional-order neutral-type fuzzy cellular bidirectional associative memory (BAM) neural networks with time-varying delays is investigated. Based on the Lyapunov–Krasovskii functional, delay-dependent sufficient conditions for solvability of the passive problem are obtained in terms of linear matrix inequalities (LMIs), which can be easily checked by using the MATLAB LMI toolbox. Finally, numerical examples are provided to show the effectiveness of the main results.

Feasible Power-Flow Solution Analysis of DC Microgrid Considering Distributed Generations Under MPPT Control

For a DC microgrid with constant power loads (CPLs), the existence of a feasible power-flow solution, which is usually difficult to analyze, is a necessary condition for the correct operation of systems. In this paper, the solvability of power-flow equation of DC microgrid with CPLs is analyzed, where a majority of distributed generations (DGs) are under MPPT control while other DGs are under droop control. At first, this paper builds a power-flow mathematical model of the DC microgrid. Secondly, three solvability conditions are proposed with the first one being applied to ensure the power-flow equation has a solution, while the other two applying to ensure the solution within the given voltage deviation. The first analytical condition ( <xref ref-type="theorem" rid="theorem2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorem 2</xref> ) is obtained by applying Brouwer fixed-point theorem. Compared with the existing results, this obtained sufficient condition is less conservative. Furthermore, the other two sufficient solvability conditions ( <xref ref-type="theorem" rid="theorem3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Theorems 3</xref> and <xref ref-type="theorem" rid="theorem4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</xref> ) can guarantee the equilibrium not only exists, but also the voltage deviation is within an acceptable range. Finally, case studies verify the correctness of the proposed theorems. The obtained conditions provide a guidance for establishing a dependable DC microgrid.

Robust Stability of DC Microgrid Under Distributed Control

Compared with AC microgrid, DC microgrid has attracted more and more attention due to their high reliability and simple control. In this paper, we analyze the existence and stability of equilibrium of DC microgrid under distributed control. Firstly, the power-flow equation of the DC microgrid with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> DGs and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> CPLs is built. On this basis, under the limitation of currents of DGs, the sufficient solvability conditions are obtained based on Brouwer’s fixed-point theorem. Secondly, we build the small-signal model to forecast the system qualitative behavior around equilibrium. The eigenvalue analysis based on inertial theorem provides the analysis basis for system stability, then we obtain the robust stability condition based on the properties of a special matrix. The obtained stability conditions provide a reference for establishing a stable DC microgrid. Lastly, simulation results verify the correctness of the proposed theorems.

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.

Unknown-input State Observers for Switching Linear Structured Systems

This work deals with the problem of designing state observers in the presence of unknown inputs for switching linear structured systems – i.e., dynamical systems which consist of a finite indexed family of linear structured systems and a switching signal indicating the active system at each time instant. Switching linear structured systems lend themselves to be described both by families of parametric state space models and by families of directed graphs, in addition to the signal ruling the switching from one mode to another. Hence, the approach adopted herein is blended. It leverages on structural notions stemmed from the geometric approach and it exploits interpretations grounded on the graph theory. The notions of switching conditioned invariant subset and switching essential output injection play a key role in the derivation of the main result, a constructive necessary and sufficient condition for solvability of the unknown-input state observation problem. The methodological discussion is illustrated by two examples.

Robust Stability Analysis of Time-Varying Delay Systems

The generalized Myshkis problem consists in analysis of robust stability of time-delay system under variation of its parameters within a set in an infinite-dimensional space. In our paper, we consider a linear system, where the delay is a bounded function under an additional restriction on its growth rate. Some sufficient conditions of solvability of the generalized Myshkis problem are expressed in the form of quadratic convex optimization task. The new approach is based on a modification of the Lyapunov-Razumikhin method.

Necessary and sufficient conditions for the well-posed solvability of a boundary value problem for a linear loaded hyperbolic equation

Problems for loaded hyperbolic equations have acquired particular relevance in connection with the study of the stability of vibrations of the wings of an aircraft loaded with masses, and in the calculation of the natural vibrations of antennas loaded with lumped capacities and self-inductions. Loaded differential equations have a number of features that must be taken into account when setting problems for these equations and creating methods for their solution. One of the features of loaded differential equations is that such equations can be undecidable without additional conditions. The main idea of the research work is to expand the class of solvable boundary value problems and develop methods that provide a numerical-analytical solution. The paper considers a boundary value problem for a linear hyperbolic equation with a mixed derivative, where the load points are set in terms of the spatial variable. By introducing unknown functions, the problem is reduced to an equivalent boundary value problem for a linear loaded hyperbolic equation of the first order. With the help of the well posed of the equivalent boundary value problem, the well posed of the original problem is established. The paper presents the necessary and sufficient conditions for the well-posedness of a periodic boundary value problem for a linear loaded hyperbolic equation with two independent variables.

A note on the solvability of homogeneous Riemann boundary problem with infinity index

Abstract In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.

Optimal control and stabilization for linear continuous‐time mean‐field systems with delay

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main contributions: (1) To the best of the authors' knowledge, the present paper is the first to establish the necessary and sufficient solvability condition for this kind of optimal control problem with input delay, and to derive the analytical form of an optimal controller through overcoming the obstacle that separation principle no longer holds for multiplicative-noise systems. (2) For the stabilization problem, under the assumption of exact observability, it is strictly proven that the system is stabilizable if and only if the algebraic Riccati equation has a unique positive definite solution.

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

Convolutions and integral equations weighted by multi-dimensional Hermite functions

In this paper, we study the solvability of a very general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are obtained. To achieve such a goal, the main technique consists in introducing eight new convolutions weighted by multi-dimensional Hermite functions and using them as convolutions somehow associated with our integral equations. In this way, two Young-type inequalities are also obtained.

Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

Optimal Control and Stabilization for Itô Systems with Input Delay

The paper considers the linear quadratic regulation (LQR) and stabilization problems for Ito stochastic systems with two input channels of which one has input delay. The underlying problem actually falls into the field of asymmetric information control because of the nonidentical measurability induced by the input delay. In contrast with single-channel single-delay problems, the challenge of the problems under study lies in the interaction between the two channels which are measurable with respect to different filtrations. The key techniques conquering such difficulty are the stochastic maximum principle and the orthogonal decomposition and reorganization technique proposed in a companion paper. The authors provide a way to solve the delayed forward backward stochastic differential equation (D-FBSDE) arising from the maximum principle. The necessary and sufficient solvability condition and the optimal controller for the LQR problem are given in terms of a new Riccati differential equation established herein. Further, the necessary and sufficient stabilization condition in the mean square sense is provided and the optimal controller is given. The idea proposed in the paper can be extended to solve related control problems for stochastic systems with multiple input channels and multiple delays.

Linear Quadratic Mean Field Games: Decentralized O(1/N)-Nash Equilibria

This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. The authors employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an O(1/N)-Nash equilibrium for the obtained decentralized strategies.

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators

We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequalityin the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap between necessary and sufficient conditions for solvability of the Dirichlet problem. For more information see https://ejde.math.txstate.edu/Volumes/2021/82/abstr.html

The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solution

The problem with non-separated multipoint-integral conditions for high-order differential equations is considered. An interval is divided into m parts, the values of a solution at the beginning points of the subintervals are considered as additional parameters, and the high-order differential equations are reduced to the Cauchy problems on the subintervals for system of differential equations with parameters. Using the solutions to these problems, new general solutions to high-order differential equations are introduced and their properties are established. Based on the general solution, non-separated multipoint-integral conditions, and continuity conditions of a solution at the interior points of the partition, the linear system of algebraic equations with respect to parameters is composed. Algorithms of the parametrization method are constructed and their convergence is proved. Sufficient conditions for the unique solvability of considered problem are set. It is shown that the solvability of boundary value problems is equivalent to the solvability of systems composed. Methods for solving boundary value problems are proposed, which are based on the construction and solving these systems.

Linear Quadratic Mean Field Social Optimization: Asymptotic Solvability and Decentralized Control

This paper studies asymptotic solvability of a linear quadratic mean field social optimization problem with controlled diffusions and indefinite state and control weights. Starting with an N-agent model, we employ a rescaling approach to derive a low-dimensional Riccati ordinary differential equation system, which characterizes a necessary and sufficient condition for asymptotic solvability. The decentralized control obtained from the mean field limit ensures a bounded optimality loss in minimizing the social cost having magnitude O(N), which implies an O(1/N) optimality loss per agent. We further quantify the efficiency gain of the social optimum with respect to the solution of the mean field game.

Asymmetric information control for stochastic systems with different intermittent observations

This paper will investigate the linear quadratic (LQ) control problem for a stochastic system with different intermittent observations. Different from traditional LQ control problems, two controllers with different information structures are considered to minimize a quadratic performance index. Specifically, the information sets available to two controllers are different and mutually exclusive (i.e., either of both is not the subset of the other one). The problem under consideration in this paper is an asymmetric information control problem. It is highlighted that previous literature on asymmetric information control mainly focused on the case when the information set is a subset of the other one. However, the case of the information sets being mutually exclusive remains less investigated to the best of our knowledge, which brings essential difficulties in finding the optimal control strategies: the structures and forms of two controllers are not clear; seeking for the solvability conditions of the optimal control problem is challenging. We summarize the contributions of this paper as follows: Firstly, by adopting the convex variational method, the necessary and sufficient solvability conditions are derived under some basic assumptions, which are relied on the forward and backward difference equations (FBSDEs); Secondly, by decoupling the FBSDEs, the control strategies are derived; Finally, the results are extended to investigate the general multiple controllers case.