Transport equations of higher order discontinuities for quasilinear hyperbolic systems

The transport equations satisfying ordinary linear differential equations of first order which govern the behaviour of higher order discontinuities for quasilinear hyperbolic systems along the rays associated with a singular surface are derived. It is shown that the transport equations depend on the Gaussian curvature of wave front.     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

Numerical method for a dynamic optimization problem arising in the modeling of a population of aerosol particles

A model coupling differential equations and a sequence of constrained optimization problems is proposed for the simulation of the evolution of a population of particles at equilibrium interacting through a common medium.The first order optimality conditions of the optimization problems relaxed with barrier functions are coupled with the differential equations into a system of differential-algebraic equations that is discretized in time with an implicit first order scheme. The resulting system of nonlinear algebraic equations is solved at each time step with an interior-point/Newton method. The Newton system is block-structured and solved with Schur complement techniques, in order to take advantage of its sparsity. Application to the dynamics of a population of organic atmospheric aerosol particles is given to illustrate the evolution of particles of different sizes. To cite this article: A. Caboussat, A. Leonard, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Constraint handling in optimization-based control of a heat-up problem using a transformation approach

An approach is presented for optimization-based control of a class of boundary-controlled parabolic partial differential equations, which are subject to state and input constraints. In order to avoid the potential numerical complexity of constrained optimization, suitably chosen asymptotic saturation functions are used to reformulate the original problem in new coordinates. The resulting unconstrained problem can then be solved with methods of unconstrained optimization, which is interesting for use in model predictive control schemes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of systems governed by hyperbolic partial differential equation with equality and inequality constraints

An optimal control problem involving nonlinear hyperbolic partial differential equations, which includes restrictions on controls and equality and inequality constraints on the terminal states, is formulated. Using this problem, a framework for obtaining (first order) necessary conditions for control problems governed by partial differential equations with equality and inequality constraints is developed.     

Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique

In this paper, a new method for solving arbitrary order ordinary differential equations and integro-differential equations of Fredholm and Volterra kind is presented. In the proposed method, these equations with separated boundary conditions are converted to a parametric optimization problem subject to algebraic constraints. Finally, control and state variables will be approximated by a Chebychev series. In this method, a new idea has been used, which offers us the ability of applying the mentioned method for almost all kinds of ordinary differential and integro-differential equations with different types of boundary conditions. The accuracy and efficiency of the proposed numerical technique have been illustrated by solving some test problems.     

Modelling the discretization error of initial value problems using the Wishart distribution

This paper presents a new discretization error quantification method for the numerical integration of ordinary differential equations. The error is modelled by using the Wishart distribution, which enables us to capture the correlation between variables. Error quantification is achieved by solving an optimization problem under the order constraints for the covariance matrices. An algorithm for the optimization problem is also established in a slightly broader context.     

Some new analysis results for a class of interface problems

Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the immersed boundary (IB) method is presented. The IB method is shown to be first‐order convergent in L ^∞ norm. Copyright © 2013 John Wiley & Sons, Ltd.     

Multilevel optimization for PDAE-constrained optimal control problems with pointwise constraints on control and state

We have developed a fully adaptive optimization environment suitable to solve complex optimal control problems restricted by partial differential algebraic equations (PDAEs) and pointwise constraints on the control [1, 2]. This contribution is devoted to the inclusion of pointwise constraints on the state within the optimization environment. To this end we first give a brief introduction into the architecture of the environment and the inclusion of pointwise constraints on the state by Moreau-Yosida regularization. Then, we test the new tool by applying it to an optimal boundary control problem for the cooling of hot glass down to room temperature, modeled by radiative heat transfer and semi-transparent boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Proof of Convergence for a Global Optimization Algorithm for Problems with Ordinary Differential Equations

A deterministic spatial branch and bound global optimization algorithm for problems with ordinary differential equations in the constraints has been developed by Papamichail and Adjiman [A rigorous global optimization algorithm for problems with ordinary differential equations. J. Glob. Optim. 24, 1–33]. In this work, it is shown that the algorithm is guaranteed to converge to the global solution. The proof is based on showing that the selection operation is bound improving and that the bounding operation is consistent. In particular, it is shown that the convex relaxation techniques used in the algorithm for the treatment of the dynamic information ensure bound improvement and consistency are achieved.     

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.     

Radiofrequency ablation planning: An application of semi-infinite modelling techniques

In radiofrequency (RF) ablation a needle-shaped probe is inserted into the patient’s body in order to heat and subsequently destroy the malignant tissue around the needle tip. The determination of the optimal probe position poses an intricate problem, as it requires the modelling of the tumour destruction depending on the attained temperature as well as the incorporation of constraints that prohibit puncturing bones or other risk structures.In this work we present a new optimization procedure that reflects both aspects adequately. We assess tumour destruction by solving the underlying system of partial differential equations using a finite element method. Next we show how the probe’s position and orientation can be optimized by gradient-based methods. Ensuring that risk structures are not harmed by the probe is easily modelled using semi-infinite constraints in the optimization problem.Techniques to reduce the semi-infinite problem to a standard nonlinear constrained optimization problem are introduced and demonstrated as a proof-of-concept on real clinical data. The results indicate the high potential of this combination of PDE-based simulation and numerical optimization for RF ablation planning.     

On an Extension of the Weak Law of Large Numbers of Kolmogorov and Feller

The robustness inequality for an optimization problem with constraints given by contractive operators is adapted to controlled stochastic differential equations. Some applications to estimation of approximation accuracy of controlled processes are discussed     

Representation of the Lagrange Multipliers for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The structure of the optimal control problem under consideration is exploited and special emphasis is laid on the representation of the Lagrange multipliers resulting from the necessary conditions for infinite optimization problems.The author thanks the referees for careful reading and helpful suggestions and comments.     

Reduction of the equations of dynamics of systems with constraints to a given structure

The problem of constructing systems of second-order ordinary differential equations, the solutions of which, with the appropriate initial conditions, satisfy given equations of the constraints, is considered. The conditions for representing the differential equations in the form of Lagrange equations of the second kind are determined. It is shown that, when the equations of the non-holonomic constraints are specified by polynomials of order no higher than two with respect to the generalized velocities, the generalized forces of a system with energy dissipation comprise the sum of the gyroscopic, potential and dissipative forces.     

A Rigorous Global Optimization Algorithm for Problems with Ordinary Differential Equations

The optimization of systems which are described by ordinary differential equations (ODEs) is often complicated by the presence of nonconvexities. A deterministic spatial branch and bound global optimization algorithm is presented in this paper for systems with ODEs in the constraints. Upper bounds for the global optimum are produced using the sequential approach for the solution of the dynamic optimization problem. The required convex relaxation of the algebraic functions is carried out using well-known global optimization techniques. A convex relaxation of the time dependent information is obtained using the concept of differential inequalities in order to construct bounds on the space of solutions of parameter dependent ODEs as well as on their second-order sensitivities. This information is then incorporated in the convex lower bounding NLP problem. The global optimization algorithm is illustrated by applying it to four case studies. These include parameter estimation problems and simple optimal control problems. The application of different underestimation schemes and branching strategies is discussed.     

Charpit System of Partial Differential Equations with Algebraic Constraints

In this paper the Charpit system of partial differential equations with algebraic constraints is considered. So, first the compatibility conditions of a system of algebraic equations and also of the Charpit system of partial differential equations are separately considered. For the combined system of equations of both types sufficient conditions for the existence of a solution are found. They lead to an algorithm for reducing the combined system to a Charpit system of partial differential equations of dimension less than the initial system and without algebraic constraints. Moreover, it is proved that this system identically satisfies the compatibility conditions if so does the initial system.     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.