Abstract. In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier-Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which smallamplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength.By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.

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This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multipliers, CPROP solves the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination. As a result, CPROP reduces the dimensionality of the optimization problem, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic and parabolic PDEs with changing parameters and nonhomogeneous terms.

This paper considers the problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented for distributed optimal control (DOC) problems in which the agent dynamics are described by a small system of stochastic differential equations (SDEs). A new set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via set-point regulation, such that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.