An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.

The aim of the work was to develop a mathematical model for the radionuclide carryover from Polissia agrolandscapes. The mathematical model was developed to reflect the radioecological situation in a limited area, the center of which is a relatively small settlement (urbanised area, village or hamlet). The conceptual scheme of the mathematical model was presented by four blocks (ecosystems or sub-landscapes), namely the block «Forest ecosystem», «Field ecosystem», «Meadow ecosystem» and «Orchard ecosystem». The model takes into account the following basic processes that affect the redistribution of 137Cs by ecosystems, depending on the ecosystem type: interception, weathering, absorption, tillage, fertilization, irrigation, organic residues decomposition, radionuclide washing out towards lower soil layers, diffusion, radionuclide fixation by soil minerals. The main differences of the created model in comparison with the existing ones are: the use of current indicators of the 137Cs soil-to-plant transition, taking into account anthropogenic effects on agroecosystems, in particular radiation countermeasures, taking into account orchard ecosystems. The mathematical model is formalized as a system of first-order linear differential equation system with fixed factors. The computer implementation of the model was performed in the mathematical package MAPLE (version 10). It was found that the 137Cs content is reducing in all ecosystems. The radionuclide removal depends on the ecosystem type, its exploitation degree, soil conditions and the complex of implemented radiation countermeasures.It was found that providing the implementation of the optimal system of radiation countermeasures and under condition of active ecosystems exploitation, all the modeled ecosystems form the following descending series of radionuclide removal: forest ecosystems — hayfields and pastures (in some cases vegetable areas) — field ecosystems (mainly cereals and legumes) — orchard ecosystems.

The class of linear differential systems with coefficient matrices which are commutative with their integrals is considered. The results on asymptotic equivalence of these systems and their distribution among linear systems are given.

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous n -dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold \mathcal{Z}\subset\mathbb{R}^n of periodic solutions satisfying \dim(\mathcal{Z}) < n. Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, x'=Mx , when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system: x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2 F_2^n(x), in \mathbb{R}^{d+2} , where \varepsilon is a small parameter, M is a (d+2)\times(d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d-m non-zero real eigenvalues.

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

This paper is devoted to linear stochastic differential systems with fractional noise (or fractional Brownian motion) input. On the basis of a convenient Markovian description of such noises, elaborated from a diffusive representation of fractional integrators previously introduced in a deterministic context, the fractional differential system is equivalently transformed into a standard (but infinite-dimensional) one, with white-noise input. Finite dimensional approximations may easily be obtained from classical discretization schemes. With this equivalent representation, the correlation function of processes described by linear fractional stochastic differential systems may be expressed from the solution of standard differential systems, which generalizes, in some way, the well-known differential Lyapunov equation, which appears when computing the covariance matrix associated with standard linear stochastic systems.

The solutions of linear fuzzy stochastic differential systems

We develop a general method for the localization of eigenvalues of matrix polynomials and functions based on the solution of matrix equations. For a broad class of equations, we formulate theorems that generalize the known properties of the Lyapunov equation. A new method for the representation of solutions of linear differential and difference systems is proposed.

An Extension of Stability Criteria for Linear and Nonlinear Time Delay Systems

The phase portraits of the planar linear differential systems are very well known. This is not the case for the phase portraits of the planar continuous piecewise linear differential systems. In this paper we classify the phase portraits of the class of planar continuous piecewise linear differential systems of the form in the Poincaré disc when , and prove the existence and uniqueness of limit cycles. Note that on the straight line these differential systems are only continuous.

Studies the optimal guaranteed control problem for a class of linear uncertain time-delay systems. The systems under consideration are described by state differential equations with time-varying unknown but norm-bounded uncertain parameters and time-varying unknown but bounded delay in states. An upper bound of the performance measured by standard quadratic function is constructed and based on the delay-dependent stabilizing approach, a method of designing a state feedback robust quadratic stabilizing control law guaranteeing this quadratic performance index be less than the constructed upper bound is derived. A procedure is given to select an optimal state feedback controller in the sense of minimizing this upper bound of the quadratic performance index. The results depend on the size of delay and are given in terms of several linear matrix inequalities.

The problems of realization and feedback stabilization of autonomous linear neutral systems are studied in the context of linear systems over commutative rings. Connections are made to the theory of retarded delay systems as well as to generalized linear systems over a field.

Cooperative output regulation problem for linear time-delay multi-agent systems under switching network

Multi-Agent Systems (MAS) can be used to model systems where the global behavior cannot be uniformly represented by standard techniques such as partial differential equations or linear systems because the system elements have their own independent behavior. This is, for instance, the case in complex systems such as daily mobility in a city for example. Depending on the system size the computing power needs for the MAS may be as big as for more traditional linear numerical systems and may need to be parallelized to fully represent real systems. Graphical Processing Units (GPU) have already proven to be an efficient support to execute large linear programs. In this paper we present the use of GPU for the execution of Sworm, a multi-scale MAS system. We show that GPU computing can be efficient in that less regular case and when the agent behavior is simple. We advocate for a wider use of the GPU in Agent Based Models in particular for multi-scale systems with work distribution between the CPU and GPU.