Finite-dimensional Problem Research Articles

Finite‐dimensional guaranteed cost sampled‐data fuzzy control of Markov jump distributed parameter systems via T–S fuzzy model

In this study, the authors are concerned with a finite-dimensional guaranteed cost sampled-data fuzzy control problem, which is addressed for a class of non-linear Markov jump parabolic partial differential equation (MJPPDE) systems. Applying the Galerkin's method to the MJPPDE system, a non-linear Markov jump ordinary differential equation (MJODE) system is derived. The resulting non-linear MJODE system can describe the dominant dynamics of the MJPPDE system, which is subsequently expressed by the Takagi–Sugeno (T–S) fuzzy model. Then, a guaranteed cost sampled-data fuzzy controller is developed to stabilise the closed-loop slow fuzzy system exponentially in the mean-square sense while providing an upper bound for the quadratic cost function. Making full use of their special features, they will be able to establish the bound on . The authors' theory enables us to guarantee the existence of the feedback controller and to judge the effectiveness of the feedback control based on the sampled-data to stabilise the given system in the sense of mean-square exponential stability.

Про проектування малочутливих прискорювально-фокусуючих систем методами практичної стійкості

Про проектування малочутливих прискорювально-фокусуючих систем методами практичної стійкості

Analysis of Integrodifference Equations with a Separable Dispersal Kernel

Integrodifference equations are a class of infinite-dimensional dynamical systems in discrete time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dispersal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite-dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.

Vectorial Slepian Functions on the Ball

Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions, in general, is to find functions that are at least optimally spatio-spectrally localized. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging. Our aim is to find band-limited vector fields that are well-localized in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimization problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localized vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localization region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localization of these functions and the accurate estimate of the Shannon number.

Uzawa-type Iterative Solution Methods for Constrained Saddle Point Problems

For finite-dimensional saddle point problem with a nonlinear monotone operator and constraints on direct variables, iterative methods are developed, which in the potential case can be viewed as preconditioned Uzawa methods or as Uzawa-block relaxation methods. Convergence conditions of the iterative methods are formulated in the form of operator inequalities connecting the operator of the problem and the preconditioning matrix. When applied to mesh problems, this allows us to construct suitable preconditioners that ensure the convergence and effective implementation of iterative methods and to obtain the admissible intervals of iterative parameters which don’t depend on mesh parameters. The presented results are based on the general theory developed by the author with co-authors in recent years.

Construction and stability of blowup solutions for a non-variational semilinear parabolic system

We consider the following parabolic system whose nonlinearity has no gradient structure:{∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:{u(x,t)∼Γ[(T−t)(1+b|x−a|2(T−t)|log⁡(T−t)|)]−(p+1)pq−1,v(x,t)∼γ[(T−t)(1+b|x−a|2(T−t)|log⁡(T−t)|)]−(q+1)pq−1, with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

Let Ω be an open region in ℝd, d ≥ 2, that is diffeomorphic to 𝔹d. Consider solving −Δu + γu = 0 on Ω with the Neumann boundary condition $\frac {\partial u}{\partial \mathbf {n}}=b\left (\cdot ,u\right ) $ over ∂Ω. The function b is a nonlinear function of u. The problem is reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems converge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over H1/2 (Ω) , and the reformulation is analyzed using tools for solving nonlinear integral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the existence and local uniqueness of a solution. For the case of three dimensions and a nonlinearity b that is given by the Stefan–Boltzmann law, we will provide an existence proof in the final section.

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation∂t2u=∂r2u+(d−1)r∂ru−(d−1)2r2sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profileu(r,t)∼Q(rλ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized ratesλ(t)∼cu(T−t)ℓγ,ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

Validated Numerics for Continuation and Bifurcation of Connecting Orbits of Maps

The present work develops validated numerical methods for analyzing continuous branches of connecting orbits—as well as their bifurcations—in one parameter families of discrete time dynamical systems. We use the method of projected boundaries to reduce the connecting orbit problem to a finite dimensional zero finding problem for a high dimensional map. We refer to this map as the connecting orbit operator. We study one parameter branches of zeros for the connecting orbit operator using existing methods of computer assisted proof know as the radii-polynomial approach and validated continuation. The local stable/unstable manifolds of the fixed points are analyzed using validated numerical methods based on the parameterization method. The validated branches of zeros found using our argument correspond to continuous families of transverse connecting orbits for the original dynamical system. Validation of a saddle node bifurcations for the connecting orbit operator correspond to a proof of existence for a tangency. We illustrate the implementation of our method for the Henon map.

Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincare map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev–Slobodeckij spaces and explicitly obtain a formal linearization of the Poincare map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.

A parallel numerical method for solving optimal control problems based on whale optimization algorithm

Development of a numerical algorithm for solving optimal control problems is reported in this article. The method is a combination of multi-staging of the original problem to a finite dimensional optimization problem and the recently proposed Whale Optimization Algorithm (WOA). The method is proposed to reduce the required number of iterations. The parallel implementation is also proposed and discussed. Numerical examples are given to check the validity and accuracy of the proposed method. Results show that method converges to the exact solution with accuracy comparable to other numerical methods. Note that the source codes are publicly available at http://www.alimirjalili.com/SourceCodes/WOA4OCP.zip.

Numerical solution of optimal control problems with explicit and implicit switches

In this article, we present a unified framework for the numerical solution of optimal control problems (OCPs) constrained by ordinary differential equations with both implicit and explicit switches. We present the problem class and qualify different types of implicitly switched systems. This classification significantly affects opportunities for solving such problems numerically. By using techniques from generalized disjunctive programming, we transform the problem into a counterpart one wherein discontinuities no longer appear implicitly. Instead, the new problem contains discrete decision variables and vanishing constraints. Recent results from the field of mixed-integer optimal control theory enable us to omit integrality constraints on variables, and allow to solve a relaxed OCP. We use a ‘first discretize, then optimize’ approach to solve the problem numerically. A direct method based on adaptive collocation is used for the discretization. The resulting finite dimensional optimization problems are mathematical programs with vanishing constraints, and we discuss numerical techniques to solve sequences of this challenging problem class. To demonstrate the efficacy and merit of our proposed approach, we investigate three benchmark problems for hybrid dynamic systems.

Blowup solutions for a reaction–diffusion system with exponential nonlinearities

We consider the following parabolic system whose nonlinearity has no gradient structure:{∂tu=Δu+epv,∂tv=μΔv+equ,u(⋅,0)=u0,v(⋅,0)=v0,p,q,μ>0, in the whole space RN. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

Constrained minimum variance control for discrete-time stochastic linear systems

We propose a computational scheme for the solution of the so-called minimum variance control problem for discrete-time stochastic linear systems subject to an explicit constraint on the 2-norm of the input (random) sequence. In our approach, we utilize a state space framework in which the minimum variance control problem is interpreted as a finite-horizon stochastic optimal control problem with incomplete state information. We show that if the set of admissible control policies for the stochastic optimal control problem consists exclusively of sequences of causal (non-anticipative) control laws that can be expressed as linear combinations of all the past and present outputs of the system together with its past inputs, then the stochastic optimal control problem can be reduced to a deterministic, finite-dimensional optimization problem. Subsequently, we show that the latter optimization problem can be associated with an equivalent convex program and in particular, a quadratically constrained quadratic program (QCQP), by means of a bilinear transformation. Finally, we present numerical simulations that illustrate the key ideas of this work.

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications. In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

Optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow

ABSTRACTThis article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.

On the Discreteness of Capacity-Achieving Distributions for Fading and Signal-Dependent Noise Channels With Amplitude-Limited Inputs

We address the problem of finding the capacity of two classes of channels with amplitude-limited inputs. The first class is frequency flat fading channels with an arbitrary (but finite support) channel gain with the channel state information available only at the receiver side; while the second one we consider is the class of additive noise channels with signal-dependent Gaussian noise. We show that for both channel models and under some regularity conditions, the capacity-achieving distribution is discrete with a finite number of mass points. Furthermore, finding the capacity-achieving distribution turns out to be a finite-dimensional optimization problem, and efficient numerical algorithms can be developed using standard optimization techniques to compute the channel capacity. We demonstrate our findings via several examples. In particular, we present an example for a block fading channel where the channel gain follows a truncated Rayleigh distribution, and two instances of signal-dependent noise that are used in the literature of magnetic recording and optical communication channels.

Linear-quadratic optimal control under non-Markovian switching

ABSTRACTWe study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.

An Analysis of Equilibria in Dense Nematic Liquid Crystals

This paper is concerned with the rigorous analysis of a recently proposed model of Nascimento et al. for describing nematic liquid crystals within the dense regime, with the orientation distribution function as the variable. A key feature of the model is that in high density regimes all nontrivial minimizers are zero on a set of positive measure so that $L^\infty$ variations cannot generally be taken about minimizers. In particular, it is unclear if the Euler--Lagrange equation is well defined, and if local minimizers satisfy it. It will be shown that there exists an analogue of the Euler--Lagrange equation that is satisfied by $L^p$ local minimizers by reducing the minimization problem to an equivalent finite-dimensional saddle-point problem, obtained by observing that on certain subsets of the domain the free-energy functional is convex so that duality methods can be applied. This analogue of the Euler--Lagrange equation is then shown to be equivalent to a vanishing variation criterion on a certain family of nonlinear curves on which the free-energy functional is sufficiently smooth. All critical points of the finite-dimensional saddle-point problem also correspond to all probability distributions where these nonlinear variations vanish. Furthermore, the analysis provides results on some qualitative phase behavior of the model.