EXAMPLE OF APPLICATION OF THE PONTRYAGIN’S MINIMUM PRINCIPLE “PMP EXTENSION”: ZERMELO PROBLEM (WITH CURRENT SPEED MORE THAN BOAT SPEED HYPOTHESIS)

We present an example of application covering several cases using the extension of the Pontryaguine minimum principle (PMP) in the case where we add a constraint on reaching a target variety at the final time: the Zermelo problem with current speed more than Boat speed hypothesis, where we consider a boat crossing a channel under a strong current and where we try to reach the opposite bank by minimizing the lateral offset or by minimizing the crossing time.

Inverse Problems with Unknown Boundary Conditions and Final Overdetermination for Time Fractional Diffusion-Wave Equations in Cylindrical Domains

Inverse problems to reconstruct a solution of a time fractional diffusion-wave equation in a cylindrical domain are studied. The equation is complemented by initial and final conditions and partly given boundary conditions. Two cases are considered: (1) full boundary data on a lateral hypersurface of the cylinder are given, but the boundary data on bases of the cylinder are specified in a neighborhood of a final time; (2) boundary data on the whole boundary of the cylinder are specified in a neighborhood of the final time, but the cylinder is either a cube or a circular cylinder. Uniqueness of solutions of the inverse problems is proved. Uniqueness for similar problems in an interval and a disk is established, too.

On the Reachability of a Feedback Controlled Leontief-Type Singular Model Involving Scheduled Production, Recycling and Non-Renewable Resources

This paper proposes and studies the reachability of a singular regular dynamic discrete Leontief-type economic model which includes production industries, recycling industries, and non-renewable products in an integrated way. The designed prefixed final state to be reached, under discussed reachability conditions, is subject to necessary additional positivity-type constraints which depend on the initial conditions and the final time for the solution to match such a final prescribed state. It is assumed that the model may be driven by both the demand and an additional correcting control in order to achieve the final targeted state in finite time. Formal sufficiency-type conditions are established for the proposed singular Leontief model to be reachable under positive feedback, correcting controls designed for appropriate demand/supply regulation. Basically, the proposed regulation scheme allows fixing a prescribed final state of economic goods stock in finite time if the model is reachable.

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

Relaxation Limit of the Aggregation Equation with Pointy Potential

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes.

Effective Multi-level Monte Carlo Methods for Stochastic Biochemical Kinetics

Stochastic mathematical models are essential for an accurate description of biochemical processes at the cellular level. The effect of random fluctuations may be significant when some species have low molecular counts. While exact stochastic simulation methods exist, they are typically expensive on systems arising in applications. Thus more effective strategies are required for simulating complex stochastic models of biochemical system. Often, the expected value of some function of the final time solution of the stochastic model is of interest. Then, the approach employing multi-level Monte Carlo methods is more efficient than the traditional techniques. In this thesis, we study multi-level Monte Carlo (MLMC) schemes for a reliable and effective simulation of stochastic models of biochemical kinetics. The advantages of these MLMC strategies are illustrated on several biochemical models arising in applications.