Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

Numerical solution of linear time delay systems using Chebyshev-tau spectral method

In this paper, a hybrid method based on method of steps and a Chebyshev-tau spectral method for solving linear time delay systems of differential equations is proposed. The method first converts the time delay system to a system of ordinary dierential equations by the method of steps and then employs Chebyshev polynomials to construct an approx- imate solution for the system. In fact, the solution of the system is expanded in terms of orthogonal Chebyshev polynomials which reduces the solution of the system to the solution of a system of algebraic equations. Also, we transform the coefficient matrix of the algebraic system to a block quasi upper triangular matrix and the latter system can be solved more efficiently than the first one. Furthermore, using orthogonal Chebyshev polynomials enables us to apply fast Fourier transform for calculating matrix-vector multiplications which makes the proposed method to be more efficient. Consistency, stability and convergence analysis of the method are provided. Numerous numerical examples are given to demonstrate efficiency and accuracy of the method. Comparisons are made with available literature

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

Optimal control of systems with memory

The “Optimal Control of Systems with memory” is a PhD project that is borne from the collaboration between the Department of Mechanical and Aerospace Engineering of Sapienza University of Rome and CNR-INM the Institute for Marine Engineering of the National Research Council of Italy (ex INSEAN). This project is part of a larger EDA (European Defence Agency) project called ETLAT: Evaluation of State of the Art Thin Line Array Technology. ETLAT is aimed at improving the scientific and technical knowledge of potential performance of current Thin Line Towed Array (TLA) technologies (element sensors and arrays) in view of Underwater Surveillance applications. A towed sonar array has been widely employed as an important tool for naval defence, ocean exploitation and ocean research. Two main operative limitations costrain the TLA design such as: a fixed immersion depth and the stabilization of its horizontal trim. The system is composed by a towed vehicle and a towed line sonar array (TLA). The two subsystems are towed by a towing cable attached to the moving boat. The role of the vehicle is to guarantee a TLA’s constant depth of navigation and the reduction of the entire system oscillations. The vehicle is also called "depressor" and its motion generates memory effects that influence the proper operation of the TLA. The dynamic of underwater towed system is affected by memory effects induced by the fluid-structure interaction, namely: vortex shedding and added damping due to the presence of a free surface in the fluid. In time domain, memory effects are represented by convolution integral between special kernel functions and the state of the system. The mathematical formulation of the underwater system, implies the use of integral-differential equations in the time domain, that requires a nonstandard optimal control strategy. The goal of this PhD work is to developed a new optimal control strategy for mechanical systems affected by memory effects and described by integral-differential equations. The innovative control method presented in this thesis, is an extension of the Pontryagin optimal solution which is normally applied to differential equations. The control is based on the variational control theory implying a feedback formulation, via model predictive control. This work introduces a novel formulation for the control of the vehicle and cable oscillations that can include in the optimal control integral terms besides the more conventional differential ones. The innovative method produces very interesting results, that show how even widely applied control methods (LQR) fail, while the present formulation exhibits the advantage of the optimal control theory based on integral-differential equations of motion

Variational Iteration Method for Partial Differential Equations with Piecewise Constant Arguments

In this paper, the variational iteration method is applied to solve the partial differential equations with piecewise constant arguments. This technique provides a sequence of functions which converges to the exact solutions of the problem and is based on the use of Lagrangemultipliers for identification of optimal value of a parameter in a functional. Employing this technique, we obtain the approximate solutions of the above mentioned equation in every interval [n, n + 1) (n = 0, 1, · · ·). Illustrative examples are given to show the efficiency of themethod

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Preconditioned fully implicit PDE solvers for monument conservation

Mathematical models for the description, in a quantitative way, of the damages induced on the monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some the asymptotic properties of the solutions are known, for a short window of time, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper a fully implicit numerical method is proposed, analyzed and numerically tested for parabolic equations of porous media type and on a systems of two PDEs that models the sulfation of marble in monuments. Due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.Comment: 26 pages, 13 figure