Numerical Optimal Control for Problems with Random Forced SPDE Constraints

Figure 1

Illustration of the optimal control and state achieved by solving Example 1, taking 𝑢(𝑥,𝑦)=exp[−64((𝑥−(1/2))²+(𝑦−(1/2))²)], 𝑁=3, 𝐾=5, and 𝛼=0.00005.

(a) Desired values

(b) Initial state

(c) Approximated solution (state)

(d) Control

(e) Adjoint

(f) Expected value

(g) Standard dev.

(h) Sparsity of global matrix 𝑛𝑧=1815980