Transversality conditions for the existence of solutions of first-order discontinuous functional differential equations

We are concerned with the existence of extremal solutions to a large class of first-order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well-known models: a generalized logistic equation or second-order problems in the presence of dry friction.

The optimality principle for second-order discrete and discrete-approximate inclusions

This paper deals with the necessary and sufficient conditions of optimality for the Mayer problem of second-order discrete and discrete-approximate inclusions. The main problem is to establish the approximation of second-order viability problems for differential inclusions with endpoint constraints. Thus, as a supplementary problem, we study the discrete approximation problem and give the optimality conditions incorporating the Euler-Lagrange inclusions and distinctive transversality conditions. Locally adjoint mappings (LAM) and equivalence theorems are the fundamental principles of achieving these optimal conditions, one of the most characteristic properties of such approaches with second-order differential inclusions that are specific to the existence of LAMs equivalence relations. Also, a discrete linear model and an example of second-order discrete inclusions in which a set-valued mapping is described by a nonlinear inequality show the applications of these results.

Variational principle for local fields

The simplest variational problems (with free, fixed boundaries, the Bolz problem) in Banach spaces are considered. Necessary conditions for a local extremum in these problems are derived. An important class of Lagrangian mechanical systems is considered – local loaded fields, for which the Lagrangian has the form of an integral functional. Necessary conditions for the action functional – the Euler-Ostrogradsky equations and transversality conditions – are obtained. The equations of the theory of elasticity and Maxwell electrodynamics are derived from the variational principle for local fields.

A Mathematical Model and Optimal Control for Listeriosis Disease from Ready-to-Eat Food Products

Ready-to-eat food (RTE) are foods that are intended by the producers for direct human consumption without the need for further preparation. The primary source of human Listeriosis is mainly through ingestion of contaminated RTE food products. Thus, implementing control strategies for Listeriosis infectious disease is vital for its management and eradication. In the present study, a deterministic model of Listeriosis disease transmission dynamics with control measures was analyzed. We assumed that humans are infected with Listeriosis either through ingestion of contaminated food products or directly with Listeria Monocytogenes in their environment. Equilibrium points of the model in the absence of control measures were determined, and their local asymptotic stability established. We formulate an optimal control problem and analytically give sufficient conditions for the optimality and the transversality conditions for the model with controls. Numerical simulations of the optimal control strategies were performed to illustrate the results. The numerical findings suggest that constant implementation of the joint optimal control measures throughout the modelling time will be more efficacious in controlling or reducing the Listeriosis disease. The results of this study can be used as baseline measures in controlling Listeriosis disease from ready-to-eat food products.

Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 < α ≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥ α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.

Optimality conditions for higher order polyhedral discrete and differential inclusions

The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs. Numerical approach is developed to solve a polyhedral problem with second order polyhedral discrete inclusions.

NNLO classical solution for Lipatov’s effective action for reggeized gluons

We consider the formalism of small-[Formula: see text] effective action for reggeized gluons[Formula: see text] and, following the approach developed in Refs. 11–17, calculate the classical gluon field to NNLO precision with fermion loops included. It is demonstrated that for each perturbative order, the self-consistency of the equations of motion is equivalent to the transversality conditions applied to the solution of the equations, these conditions allow to construct the general recursive scheme for the solution’s calculation. The one fermion loop contribution to the classical solutions and application of the obtained results are also discussed.

Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratioR0is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.