A02
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- Global classical solutions Ideal gas Bi-directional flow Transsonic flow (1)
- adjoint calculus (1)
- hyperbolic systems of conservation laws (1)
- hyperbolic systems of conservation laws, shock curves, generalized riemann problem, optimal control, variational calculus (1)
- networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control (1)
- optimal control (1)
- optimal control, scalar conservation law, differentiability, adjoint state, shock sensitivity (1)
- optimal control, scalar conservation law, network (1)
We derive an adjoint gradient representation for the objective functional of an optimal control problem of the unique entropy solution to the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation or balance laws.
The GRP is an initial value problem where the initial state is piecewise C^1 with exactly one discontinuity. In a recent work, differentiability properties of the solution operator of the GRP were derived. In particular, the differentiability of a class of tracking-type functionals was shown. We build on these results to derive an adjoint gradient representation for such functionals.
The adjoint problem is a system of linear transport equations with a discontinuous coefficient and discontinuous terminal state. We prove the existence and uniqueness of a piecewise Lipschitz continuous solution to the adjoint problem which satisfies suitable interior boundary conditions along all shock curves. The adjoint state generally contains infinitely many discontinuities but its total variation remains uniformly bounded in time.
We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels.
It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones.
We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls.
We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable (C^1) states left and right of it. The control consists of the C^1 parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon T>0 to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small.
Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to C^0. In physical coordinates, this implies that the shock curves in C^1 and the states between the shocks in the topology of C^0 depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals.
We study the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal boundary control problems governed by entropy solutions of conservation laws. As boundary control we consider piecewise continuously differentiable controls with possible discontinuities at switching times, where the smooth parts as well as the switching times serve as controls. The derivative of tracking-type objective functionals with respect to the smooth controls and the switching times can then be represented by an adjoint-based formula. The main difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and boundary conditions that lead in general to discontinuous adjoints. Moreover, the solution of the adjoint equation is non-unique and the so-called reversible solution leads to the correct adjoint-based derivative representation.
We study discrete adjoint schemes of monotone difference schemes in conservation form such as Engquist-Osher or Lax-Friedrichs scheme. We also allow that the state is computed by another numerical scheme satisfying certain convergence properties. We proof convergence results of the discrete adjoint to the reversible solution.
The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the weak^* topology of L^\infty if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.
n PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however,and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context.In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.
We analyze the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal control problems governed by entropy solutions of conservation laws. The difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and discontinuous end data. We derive the discrete adjoint scheme for monotone difference schemes in conservation form. It is known that convergence of the discrete adjoint can only be expected if the numerical scheme has viscosity of order O(h^\alpha) with appropriate 0 < \alpha < 1, which leads to quite viscous shock profiles. We show that by a slight modification of the end data of the discrete adjoint scheme convergence to the correct reversible solution can be obtained also for numerical schemes with viscosity of order O(h) and with sharp shock resolution. The theoretical findings are confirmed by numerical results.
In this paper we analyze the optimal control of initial-boundary value problems for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Hereby, we suppose that the initial and the boundary data switch between different C¹-functions at certain switching points, where the C¹ -functions and the switching points are considered as the control. For a class of cost functionals, we prove first order necessary optimality conditions for the corresponding optimal control problem with state constraints. Furthermore, we use a Moreau-Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality conditions for the regularized problems and finally prove convergence to the solution of the optimal control problem with state 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.
