Furtivity and Masking Problems in Time-Dependent Electromagnetic Obstacle Scattering

Summary form only given. We consider furtivity and masking problems in time dependent three dimensional electromagnetic obstacle scattering. That is we propose a criterion based on a merit function to "minimize" or to "mask" the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and with respect to this criterion we derive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time dependent acoustic obstacle scattering. We propose mathematical models of the furtivity and masking time dependent three dimensional electromagnetic scattering problems that consist in optimal control problems for the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover the first order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series. We assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Really impressive speed up factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website: http://www.econ.unian.it/reechioni/w10/. Finally further developments of the ideas presented here will be discussed.

In this paper, we consider one optimal control problem with a multipoint quality functional described by a system of nonlinear hyperbolic equations with Goursat boundary conditions. Using a modified version of the increment method, various necessary first-order optimality conditions such as the Pontryagin maximum principle and the linearized maximum condition are proved.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

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 new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

Singular perturbation problems for systems of partial differential equations of elliptic type

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

Boundary value problems are considered for a system of stationary partial differential equations that stems from the models describing the implantation of impurities into semiconductor devices and is similar to the equations of carrier transport. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.