We consider a controlled second order
stochastic partial differential equation (SPDE) with Levy noise. To solve this system numerically,
we apply a Galerkin scheme leading to a sequence of
ordinary SDEs of large order. To reduce the high dimension we use balanced truncation.
A. C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control 6. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2005.
[2]
P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM J. Control Optim., 49 (2011), 686-711.
[3]
R. F. Curtain, Stability of Stochastic Partial Differential Equation, J. Math. Anal. Appl., 79 (1981), 352-369.
[4]
T. Damm, Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Information Sciences 297, Berlin: Springer, 2004.
[5]
W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Aust. Math. Soc., 54 (1996), 79-85.
[6]
E. Hausenblas, Approximation for Semilinear Stochastic Evolution Equations, Potential Anal., 18 (2003), 141-186.
[7]
A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. A 2009, 465 (2009), 649-667.
[8]
B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Autom. Control, 26 (1981), 17-32.
[9]
S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An evolution equation approach, Encyclopedia of Mathematics and Its Applications 113, Cambridge: Cambridge University Press, 2007.
[10]
A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems, SIAM Rev., 23 (1981), 25-52.
[11]
M. Redmann and P. Benner, Model Reduction for Stochastic Systems, Stoch PDE: Anal Comp, 3(3) (2015), 291-338.