Analysis Of Optimal Control Problem Research Articles

Numerical analysis of optimal control problems governed by fourth‐order linear elliptic equations using the Hessian discretization method

AbstractThis article focuses on the optimal control problems governed by fourth‐order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretization method (HDM). The HDM is an abstract framework that enables the convergence analysis of several numerical methods such as the conforming finite element methods, the Adini and Morley nonconforming finite element methods (ncFEMs), the method based on gradient recovery (GR) operators, and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint, and control variables. A companion operator for the GR method with specific property is designed. The article concludes with numerical results that illustrate the theoretical convergence rates for GR method, Adini ncFEM, and FVM.

Оптимальное мультипликативное управление полулинейным параболическим уравнением

An analysis of optimal control problems for a nonlinear parabolic initial-boundary value problem that models the dynamics of the collective behavior of a bacterial community is presented. Estimates for the solution of the initial-boundary value problem are obtained, the solvability of control problems is proved, and optimality conditions are derived. The weak "bang-bang" principle is set for the problem with the final observation.

Asymptotic analysis of optimal control problems on the semiaxes for Carathéodory differential inclusions with fast oscillating coefficients

We consider an optimal control problem for a differential inclusion of the Carathéodory type affine with respect to the control with a coercive cost functional on a semiaxis and with fast oscillating time-dependent coefficients. We prove that, when the small parameter converges to zero, the solution to this problem tends to some solution of the optimal control problem with averaged coefficients, where the averaging we understand in the sense of the Kuratowski upper limit.

Stability analysis of optimal control problems with time-dependent costs

We present stability conditions for deterministic time-varying nonlinear discrete-time systems whose inputs aim to minimize an infinite-horizon time-dependent cost. Global asymptotic and exponential stability properties for general attractors are established. This work covers and generalizes the related results on discounted optimal control problems to more general systems and cost functions.

A new design and analysis of optimal control problems arising from COVID‐19 outbreak

In this manuscript, we append the hospitalization, diagnosed and isolation compartments to the classic SEIR model to design a new COVID‐19 epidemic model. We further subdivide the isolation compartment into asymptomatic infected and symptomatic infected compartments. For validity of the purposed model, we prove the existence of a unique solution and prove the positivity and boundedness of the solution. To study disease dynamics, we compute equilibrium points and the reproduction number . We also investigate the local and global stabilities at both of the equilibrium points. Sensitivity analysis will be performed to observe the effect of transmission parameters on . For optimal control analysis, we design two different optimal control problems by taking different optimal control approaches. Firstly, we add an isolation compartment in the newly designed model, and secondly, three parameters describing non‐pharmaceutical behaviors such as educating people to take precautionary measures, providing intensive medical care with medication, and utilizing resources by government are added in the model. We set up optimality conditions by using Pontryagin's maximum principle and develop computing algorithms to solve the conditions numerically. At the end, numerical solutions will be displayed graphically with discussion.

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

The method of penalty functions in the analysis of optimal control problems of Navier — Stokes evolutionary systems with a spatial variable in a network-like domain

The article considers the Navier — Stokes evolutionary differential system used in the mathematical description of the evolutionary processes of transportation of various types of liquids through network or main pipelines. The Navier—Stokes system is considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional networklike domains. These domains are a totality of a finite number of mutually non-intersecting subdomains connected to each other by parts of the surfaces of their boundaries like a graph (in applications these are the places of branching of pipelines). Two main questions of analysis are discussed: the weak solvability of the initial boundary value problem of the Navier — Stokes system and the optimal control of this system. The main method of research of weak solutions is the semidigitization of the input system by a time variable, that is the reduction of a differential system to a differential-difference system, and using a priori estimates for weak solutions of boundary value problems to prove the theorem of the existence of a solution of the input differential system. For the optimal control problem a minimizing functional (the penalty function) and a family of the approximate functional with parameters that characterize the “penalty” for failure to fulfill the equations of state of the system are introduced. At the same time, a special Hilbert space is created, the elements of which are pairs of functions that describe the state of the system and controlling actions. The convergence of the sequence of such functions to the optimal state of the system and its corresponding optimal control is proved. The latter essentially widen the possibilities of analysis of stationary and nonstationary network-like processes of hydrodynamics and optimal control of these processesd.

Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities

The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-hemivariational inequalities, which extends the recent results established by Han-Migórski-Sofonea [12, Theorem 9] and Migórski-Bai [23, Theorem 5]. Then, an optimal control problem driven by a (DVHVI) is studied and an existence theorem is delivered. Finally, we explore the sensitivity properties of the optimal control problem under the consideration depending on the initial data ξ∈H and some further parameter λ∈E.

Оптимальное управление уравнениями радиационного теплообмена для многокомпонентных сред

An analysis of optimal control problems for nonlinear elliptic equations modeling complex heat transfer with Fresnel conjugation conditions on the discontinuity surfaces of the refractive index is presented. Conditions for the solvability of extremal problems and the nondegeneracy of the optimality system are obtained. For the control problem with boundary observation, the bang-bang property is set.

Sensitivity Analysis of Optimal Control Problems Governed by Nonlinear Hilfer Fractional Evolution Inclusions

This article studies sensitivity properties of optimal control problems that are governed by nonlinear Hilfer fractional evolution inclusions (NHFEIs) in Hilbert spaces, where the initial state $$\xi $$ is not the classical Cauchy, but is the Riemann–Liouville integral. First, we obtain the nonemptiness and the compactness properties of mild solution sets $$\mathbb {S}(\xi )$$ for NHFEIs, and also establish an extension Filippov’s theorem. Then we obtain the continuity and upper semicontinuity of optimal control problems connected with NHFEIs depending on a initial state $$\xi $$ as well as a parameter $$\lambda $$ . Finally, An illustrating example is given.

Optimal control of a differential-difference parabolic system with distributed parameters on the graph

In the paper be considered the problem of optimal control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. Particular attention is given to the connection of the differential-differential system with the evolutionary differential system and the search conditions in which the properties of the differential system are preserved. This connection establishes a universal method of semi- digitization by temporal variable for differential system, providing an effective tool in finding conditions of uniqueness solvability and continuity on the initial data for the differential- differential system. A priori estimates of the norms of a weak solution of differential- differential system give an opportunity to establish not only the solvability of this system but also the existence of a weak solution of the evolutionary differential system. For the differential-difference system analysis of the optimal control problem is presented, containing natural in that cases a additional study of the problem with a time lag. This essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — defining ratios that determine optimal control or the set optimal controls. The work shows courses to transfer the results in case of analysis of optimal control problems in the class of functions with bearer in network-like domains. The transition from an evolutionary differential system to a differential-difference system was a natural step in the study of applied problems of the theory of the transfer of solid mediums. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.

Устойчивость слабого решения гиперболической системы с распределенными параметрами на графе

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Optimal Control and Stability Analysis of Nonlinear Control-Affine Systems

In this paper we will discuss the problems of optimal control and stabilization for nonlinear control-affine systems of the form where and vector functions A(x), B(x) are assumed to be smooth in the domain D ⊂ ℝ n , 0 ∈ D, A(0) = 0. We give an in-depth analysis of optimal control problems for nonlinear dynamical systems and then consider for the above systems the problem of synthesis of continuous control u = u(x), u(0) = 0, that stabilizes the system at the equilibrium point (x, u) = (0, 0). The solution to the problem is based on the transformation of the system to canonical form, and on nonlinear stabilization of the system.

Point control of a differential-difference system with distributed parameters on the graph

The article considers the problem of point control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. The differential- difference system is closely related to the evolutionary differential system and moreover the properties of the differential system are preserved. This connection is established by the universal method of semi-discretization in a time variable for a differential system, which provides an effective tool in order to find conditions for unique solvability and continuity on the initial data for the differential-difference system. For this differential-difference system, a special case of the optimal control problem is studied: the problem of point control action on the controlled differential-difference system is considered by the control, concentrated at all internal nodes of the graph. At the same time, the restrictive set of permissible controls is set by the means of conditions depending on the nature of the applied tasks. In this case, the controls are concentrated at the end points of the edges adjacent to each inner node of the graph. This is a characteristic feature of the study presented, quite often used in practice when building a mechanism for managing the processes of transportation of different kinds of masses over network media. The study essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — obtained ratios that determine optimal point control. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on the graph, which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Analysis of Optimal Control Problems for Hybrid Systems with One State Variable

We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state space to another. Assuming the state and control space are one-dimensional, we provide structural results of the candidate optimal solutions. In particular, we show that candidate optimal state trajectories are monotonic. Further, we prove a theorem on absence of limit cycles, generalizing the Bendixson criterion to a hybrid context. Using these results, we derive necessary and sufficient conditions for the existence of tipping points for this class of problems. A tipping point is an initial condition starting from which there exist multiple candidate optimal solutions with different qualitative behaviors. Next, we specialize these results for piecewise linear quadratic models to obtain an analytic characterization of tipping points in terms of the model parameters.

Analysis of Optimal Control Problems for Hybrid Systems with One State Variable

We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state sp...

Numerical methods for construction of value functions in optimal control problems with infinite horizon

The article is devoted to the analysis of optimal control problems with infinite time horizon. These problems arise in economic growth models and in stabilization problems for dynamic systems. The problem peculiarity is a quality functional with an unbounded integrand which is discounted by an exponential index. The problem is reduced to an equivalent optimal control problem with the stationary value function. It is shown that the value function is the generalized minimax solution of the corresponding Hamilton-Jacobi equation. The boundary condition for the stationary value function is replaced by the property of the Hölder continuity and the sublinear growth condition. A backward procedure on infinite time horizon is proposed for construction of the value function. This procedure approximates the value function as the generalized minimax solution of the stationary Hamilton-Jacobi equation. Its convergence is based on the contraction mapping method defined on the family of uniformly bounded and Holder continuous functions. After the special change of variables the procedure is realized in numerical finite difference schemes on strongly invariant compact sets for optimal control problems and differential games.

On group analysis of optimal control problems in economic growth models

The optimal control problems in economic growth theory are analyzed by considering the Pontryagin's maximum principle for both current and present value Hamiltonian functions based on the theory of Lie groups. As a result of these necessary conditions, two coupled first-order differential equations are obtained for two different economic growth models. The first integrals and the analytical solutions (closed-form solutions) of two different economic growth models are analyzed via the group theory including Lie point symmetries, Jacobi last multiplier, Prelle-Singer method, $ \lambda $-symmetry and the mathematical relations among them.

Analysis of optimal control problem of HIV-1 model of engineered virus

Analysis of optimal control problem of HIV-1 model of engineered virus

Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution

An optimal control problem of contaminated underground water is considered. The spatio-temporal objective takes into account the economic trade off between the pollutant use –for instance fertilizer– and the cleaning costs. It is constrained by a hydrogeological model for the spread of the pollution in the aquifer. We consider a broad range of reaction kinetics. The aim of the paper is two-fold. On the one hand, we rigorously derive, by asymptotic analysis, the effective optimal control problem for contaminant species that are slightly concentrated in the aquifer. On the other hand, the mathematical analysis of the optimal control problems is performed and we prove in particular that the latter effective problem is well-posed. Furthermore, a stability property of the optimal control process is provided: any optimal solution of the properly scaled problem tends to the optimal solution of the effective problem as the characteristic pollutant concentration decreases.