Multi-agent Optimal Control Problems and Variational Inequality Based Reformulations

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control. The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods. Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

A gradient method for solving the optimal control problem for the system of telegraph equations

The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating of singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without delay. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. In future studies, it is also planned to construct an asymptotic representation of the solution of a control system with delays and degeneracies. Keywords: optimal control system, systems of differential equations with delay, asymptotic solution, turning point.

System of fractional differential algebraic equations with applications

Index-analysis for a method of lines discretising multirate partial differential algebraic equations

The optimization problem consists in finding an element that minimizes or maximizes the goal (or cost) functional on the set of admissible controls. The paper is devoted to the optimization problem for a pseudoparabolic equation with control in coefficients of the variational equality as well as on the right-hand side. We consider optimal control problem (parabolic initial boundary value problem) of a viscoelastic plate. The bending of a plate is described by means of the Kirchhoff model. We assume that a nonhomogeneous and anisotropic viscoelastic plate occupying a domain Ω × (—H, H) of the space ℝ3 is loaded be a transversal distributed force perpendicular to the middle plane of the plate. The viscoelastic reinforced plate with welded bars rests on a unilateral elastic foundation. The role of the control variables is played by the thickness of the viscoelastic plate and the stiffness characteristics of the elastic Winkler medium, respectively. We consider the desired moments field (the bending moments and torque) in time T as the cost functional. The control variables have to belong to a set of Lipschitz-continuous functions and to a set of continuously differentiable functions, respectively. The state problem is modelled by a pseudoparabolic variational equality, where the control variables influence both the coefficients of the linear strictly monotone operator and the Winkler functional. We derive there some apriori estimates of the state function which are useful for the problem of the existence of the optimal thickness function and the Winkler modulus, respectively, and for unicity of the optimal controls. On the basis of a general existence theorem for a class of pseudoparabolic optimization problems with the state variational equation, we prove the existence of at least one solution of the optimal control problem.

A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques.Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear.Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

The first attempt to improve battery performance through model-based design optimization was made by W. Tiedemann and J. Newman in 19751. They varied the electrode thickness and porosity and then observed the effects on the effective capacity. Newman later applied the reaction-zone model to maximize the specific energy of the system using the same varying and observing method2. V. Ramadesigan et al.3developed optimal spatially varying porosity for porous electrodes by considering ohmic resistance model in the electrode. They minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using graded electrode to further improve the performace. Since the introduction of the pseudo-2D (P2D) model by the Newman group, many efforts have been made in utilizing the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method4. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis5 incorporating mechanical stress-strain relationship. S. De and his coworkers6 used a reformulated model, to perform simultaneous optimization of multiple design parameters. N. Xue et al. 7 applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan8used a gradient-free direct search method to maximize the specific energy. Z. Du et al. studies the effects of various design parameters on thick electrode performance9. Most of the aforementioned optimization research, used the sequential optimization approach, also known as CVP. CVP takes the differential algebraic equation (DAE) system describing the battery physics, applies a certain nonlinear programming (NLP) solver to them, discretizes only the control variables, and solves the model. The only exception being the effort by Golmon et al which used an indirect approach. Note that as of today, global optimization cannot be guaranteed when CVP type approaches are used with P2D type models. In general optimization approaches can be classified as indirect and direct approach. In the direct approach there are two main categories – sequential and simultaneous approach. The simultaneous approach10, on the other hand, discretizes both the control variables and the design variables before solving the problem. When used for optimization, the DAE system will only be solved once at the optimal point, compared to the repeated numerical integration needed for sequential optimization. By using high order discretization scheme on both the state and the design variables, simultaneous optimization approach results in faster determination of the optimum with fewer time steps. Furthermore, this approach offers more flexibility over constraints on the state variables. Since the state variables are also discretized, it is possible to apply equality/inequality constraints on their internal values. In this work, we will first apply the simultaneous optimization approach to a resistance model similar to the one mentioned earlier3, and compare its performance with the sequential approach. We will also explore the application of the same optimization approach to the P2D model, and identify the challenges. It is viewed as a controversial topic in the literature wherein different authors disagree on the relevance of graded porosity for improved performance8. By using a simultaneous optimization approach and guaranteeing global optimization for convex problem statements, we hope to provide additional insights. Acknowledgements The authors are thankful for the financial support by the Clean Energy Institute at the University of Washington and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).

This paper proposes notions and a tool for passivity properties of non-homogeneous switched Differential Algebraic Equation (DAE) systems and their relationships with stability and control design. Motivated by the lack of results on input-output analysis (such as passivity) for switched DAE systems and their interconnections, we propose to model non-homogeneous switched DAE systems as a class of hybrid systems, modeled here as hybrid DAE systems with linear flows. Passivity and its variations are defined for switched DAE systems and methods relying on storage functions are proposed. The main contributions of this paper are: 1) passivity and detectability concepts for switched DAE systems, 2) links of the aforementioned passivity and detectability properties to stabilization via static output-feedback. Our results are illustrated in a power system, namely, the DC-DC boost converter, whose model involves DAEs and requires feedback control.