While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.
In this paper, optimal control problems subject to a nonlinear scalar conservation law are
studied. Such optimal control problems are challenging both at the continuous and at the discrete
level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore
discretization of the underlying optimal control problem should be designed with care. Here the
discretize-then-optimize approach is employed where first the full discretization of the objective
function as well as the underlying PDE is considered. Then, the derivative of the reduced objective
is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta
(TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods,
also called strong stability preserving (SSP), are originally designed to preserve total variation of
the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to
ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is
too strong. Also approximation properties that the discrete adjoint inherits from the discretization
of the state equation are studied. Moreover order conditions are derived. In addition, optimal
choices with respect to CFL constant are discussed and numerical experiments are presented.
In this paper we study the structure of solutions of the one dimensional weighted total
variation regularisation problem, motivated by its application in signal recovery tasks. We study
in depth the relationship between the weight function and the creation of new discontinuities in
the solution. A partial semigroup property relating the weight function and the solution is shown
and analytic solutions for simply data functions are computed. We prove that the weighted total
variation minimisation problem is well-posed even in the case of vanishing weight function, despite
the lack of coercivity. This is based on the fact that the total variation of the solution is bounded
by the total variation of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two problems can produce
completely different solutions even for very simple data functions.
This paper is concerned with the distributed optimal control of a time-discrete Cahn–
Hilliard/Navier–Stokes system with variable densities. It focuses on the double-obstacle potential
which yields an optimal control problem for a family of coupled systems in each time instant of a
variational inequality of fourth order and the Navier–Stokes equation. By proposing a suitable time-
discretization, energy estimates are proved and the existence of solutions to the primal system and of
optimal controls is established for the original problem as well as for a family of regularized problems.
The latter correspond to Moreau–Yosida type approximations of the double-obstacle potential. The
consistency of these approximations is shown and first order optimality conditions for the regularized
problems are derived. Through a limit process with respect to the regularization parameter, a
stationarity system for the original problem is established. The resulting system corresponds to a
function space version of C-stationarity which is a special notion of stationarity for MPECs.
We address the problem of optimally placing sensor networks for convection-diffusion
processes where the convective part is perturbed. The problem is formulated as an optimal control
problem where the integral Riccati equation is a constraint and the design variables are sensor
locations. The objective functional involves a term associated to the trace of the solution to the
Riccati equation and a term given by a constrained optimization problem for the directional derivative
of the previous quantity over a set of admissible perturbations. The paper addresses the existence
of the derivative with respect to the convective part of the solution to the Riccati equation, the
well-posedness of the optimization problem and finalizes with a range of numerical tests.
Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.
Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.
Using a standard first-order optimality condition for nonsmooth optimization prob-
lems, a general framework for a descent method is developed. This setting is applied to
a class of mathematical programs with equilibrium constraints in function space from
which a new algorithm is derived. Global convergence of the algorithm is demonstrated
in function space and the results are then illustrated by numerical experiments.
The directional differentiability of the solution mapping for a class of variational inequali-
ties of the second kind inspired by applications in fluid mechanics and moving free boundary
problems is investigated. The result is particularly relevant for the model predictive control
or optimal control of such variational inequalities in that it can be used to derive stationarity
conditions and efficient numerical methods.
A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples.