Penalized optimal controls for the heat equation converge to the hard-constrained solution at explicit rates O(alpha to the minus theta), with sharp O(1/alpha) under stronger assumptions on the terminal mismatch.
Boundary control of semilinear elliptic equations with pointwise state constraints.SIAM Journal on Control and Optimization, 31(4):993–1006, 1993
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Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation
Penalized optimal controls for the heat equation converge to the hard-constrained solution at explicit rates O(alpha to the minus theta), with sharp O(1/alpha) under stronger assumptions on the terminal mismatch.