Infinite horizon backward stochastic Volterra integral equations and discounted control problems

Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted L2space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted L2-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin’s maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.

Multi-objective aerodynamic optimization of flying-wing configuration based on adjoint method

In order to solve the multi-objective multi-constraint design in aerodynamic design of flying wing, the aerodynamic optimization design based on the adjoint method is studied. In terms of the principle of the adjoint equation, the boundary conditions and the gradient equations are derived. The Navier-Stokes equations and adjoint aerodynamic optimization design method are adopted, the optimization design of the transonic drag reduction for the two different aspect ratio of the flying wing configurations is carried out. The results of the optimization design are as follows: Under the condition of satisfying the aerodynamic and geometric constraints, the transonic shock resistance of the flying wing is weakened to a great extent, which proves that the developed method has high optimization efficiency and good optimization effect in the multi-objective multi-constraint aerodynamic design of the flying wing.

3D general-measure inversion of crosswell EM data using a direct solver

We present a three-dimensional (3D) general-measure inversion scheme of crosswell electromagnetic (EM) data in the frequency domain with a direct forward solver. In the forward problem, we discretised the EM Helmholtz equation by the staggered-grid finite difference (SGFD) scheme and solved it using the Intel MKL PARDISO direct solver. By applying a direct solver, we simultaneously solved the multisource forward problems at a given frequency. In the inversion, we integrated a general measure of data misfit and model constraints with linearised least-squares inversion. We reconstructed a model with blocky features by selecting the appropriate measure parameters and model constraints. We used the adjoint equation method to explicitly calculate the Jacobian matrix, which facilitated the determination of an appropriate initial value for the regularisation coefficient in the objective function. We illustrated the inversion scheme using synthetic crosswell EM data with a general measure, the L2 norm, and, specifically, two mixed norms.

Hypervolume scalarization for shape optimization to improve reliability and cost of ceramic components

In engineering applications one often has to trade-off among several objectives as, for example, the mechanical stability of a component, its efficiency, its weight and its cost. We consider a biobjective shape optimization problem maximizing the mechanical stability of a ceramic component under tensile load while minimizing its volume. Stability is thereby modeled using a Weibull-type formulation of the probability of failure under external loads. The PDE formulation of the mechanical state equation is discretized by a finite element method on a regular grid. To solve the discretized biobjective shape optimization problem we suggest a hypervolume scalarization, with which also unsupported efficient solutions can be determined without adding constraints to the problem formulation. FurthIn this section, general properties of the hypervolumeermore, maximizing the dominated hypervolume supports the decision maker in identifying compromise solutions. We investigate the relation of the hypervolume scalarization to the weighted sum scalarization and to direct multiobjective descent methods. Since gradient information can be efficiently obtained by solving the adjoint equation, the scalarized problem can be solved by a gradient ascent algorithm. We evaluate our approach on a 2 D test case representing a straight joint under tensile load.

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Special cases of critical linear difference equations

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

Optimal control of the two-dimensional Vlasov-Maxwell system

The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.

Feedback control of chaotic systems using multiple shooting shadowing and application to Kuramoto–Sivashinsky equation

We propose an iterative method to evaluate the feedback control kernel of a chaotic system directly from the system’s attractor. Such kernels are currently computed using standard linear optimal control theory, known as linear quadratic regulator theory. This is however applicable only to linear systems, which are obtained by linearizing the system governing equations around a target state. In the present paper, we employ the preconditioned multiple shooting shadowing (PMSS) algorithm to compute the kernel directly from the nonlinear dynamics, thereby bypassing the linear approximation. Using the adjoint version of the PMSS algorithm, we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto–Sivashinsky equation. We compare the computed kernel with that produced by the standard linear quadratic regulator algorithm and note similarities and differences. Both kernels are stabilizing, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.