A new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise.

We revisit the American put and call option valuation problems. We derive analytical formulas for the option prices and approximate ordinary differential equations for the optimal exercise boundaries. Numerical simulations yield accurate option prices and comparable computational speeds when benchmarked against the binomial method for calculating option prices. Our approach is based on the Mellin transform and an adaptation of the Kármán–Pohlausen technique for boundary layers in fluid mechanics.

This work models and analyses the dynamics of a general spring-mass-damper system that is in frictional contact with its support, taking into account frictional heat generation and a reactive obstacle. Friction, heat generation and contact are modelled with subdifferentials of, possibly non-convex, potential functions. The model consists of a non-linear system of first-order differential inclusions for the position, velocity and temperature of the mass. The existence of a global solution is established and additional assumptions yield its uniqueness. Nine examples of conditions arising in applications, for which the analysis results are valid, are presented.

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) injection/suction of a viscous fluid at the walls. We obtain an asymptotic expansion of the solution of the Navier–Stokes equations in the limit when the amplitude of normal displacements of fluid particles near the walls is much smaller than both the width of the channel and the thickness of the Stokes layer. It is shown that the steady part of the flow in this problem is much stronger than the steady flow produced by vibrations of impermeable boundaries. Another interesting feature of this problem is that the direction of the steady flow is opposite to what one would expect if the flow was produced by vibrations of impermeable walls.

Reduced ordinary differential equation (ODE) models arising from a high-order lubrication system and describing coarsening dynamics of droplets in nanometric polymer film interacting on a hydrophobically coated solid substrate in the presence of large slippage at the liquid/solid interface are analysed. In the limiting case of infinite slip length corresponding in applications to free films, a collision/absorption model then arises and is solved explicitly. The exact coarsening law is derived for it analytically and confirmed numerically. Existence of a threshold for the decay of initial distributions of droplet distances at infinity at which the coarsening rates switch from algebraic to exponential ones is shown.

In this paper, we study the regularity of a weak solution for a coupled system derived from a microwave-heating model. The main feature of this model is that electric conductivity in the electromagnetic field is assumed to be temperature dependent. It is shown that the weak solution of the coupled system possesses some regularity under certain conditions. In particular, it is shown that the temperature is Hölder continuous, even if electric conductivity has a jump discontinuity with respect to the temperature change. The main idea in the proof is based on an estimate for a linear degenerate system in Campanato space. As an application, the regularity result for the coupled system is used to derive the necessary condition for an optimal control problem arising in microwave heating processes.

We review the mathematical analysis of some uniaxial, liquid crystal phases. Firstly, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models. It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modelled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and obtain analytical results.