Optimal Control of the Multiphase Stefan Problem

Abstract

We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the L 2 -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove wellposedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform L ∞ bound, and W 1,1 2 -energy estimate for the discrete multiphase Stefan problem.Key words: Inverse multiphase Stefan problem, quasilinear parabolic PDE with discontinuous coefficent, optimal control, Sobolev spaces, method of finite differences, discrete optimal control problem, energy estimate, embedding theorems, weak compactness, convergence in functional, convergence in control.AMS subject classifications: 35R30, 35R35, 35K20, 35Q93, 65M06, 65M12, 65M32, 65N21. * This research is funded by NSF grant #1359074Inverse Multiphase Stefan Problem (IMSP). Find the functions u(x, t), ξ j (t), j = 1, J, and the boundary heat flux g(t) satisfying (1)-(6),(8).Motivation for the IMSP arose from the modeling of bioengineering problems on the laser ablation of biological tissues through a multiphase Stefan problem (1)-(6). Laser ablation creates three phases -solid (skin), fluid (melted skin) and gas (evaporated skin). Free boundaries ξ 1 (t) and ξ 2 (t) are measuring ablation depth separating solid/fluid and fluid/air regions at the moment t. GR±ε J (g), ε > 0. Then lim ε→0 J * (ε) = J * = lim ε→0 J * (−ε).Proof. The proof of this lemma is very similar to the analogous lemma from [1]. If 0 < ε 1 < ε 2 , then

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

“…Having estimates (4.1),(4.2) and compactness Theorem 4.3, the completion of the proof of Theorems 2.1 and 2.2 coincides with the proof given in [5], through compactness arguments and proving weak continuity of cost functional J , and verification of the conditions of Theorem 3.2 (see Lemmas A, B and C in [5]).…”

“…We also have that the last integral tends to 0 due to absolute continuity of the integral. Using the convergence properties of the interpolations, due to weak convergence of {v τ }, equivalence of {v τ } and {v τ }, and uniform convergence of {ψ τ x }, passing to the limit as n → +∞ we get: Sinceb(x, t) andb 0 (x) are both of type B, and by use of Mazur's lemma, we deduce as in [5] thatb(x, t) = B(x, t, v(x, t))b 0 (x) = B(x, 0, Φ(x)) a.e on D and (0, ℓ)…”

“…Proof. Having estimates (4.1),(4.2), the proof is pursued similar to the proof of Theorem 5 in [5]. By the definitions of the interpolations in (1.19) we have that there is a subsequence of {v τ } that converges weakly in W 1,1 2 (D) to some function v ∈ W 1,1 2 (D)∩L ∞ (D), strongly in L 2 (D), and a further subsequence that converges to v pointwise almost everywhere in D. We also have equivalence of {v τ } and {v τ } in W 1,0 2 (D), and equivalence of {v τ } and {ṽ} in L 2 (D).…”

Identification of parameters in systems biology

“…Some of the variation in the chart can be attributed to error accumulation and noise. Table 5 demonstrates the final values of 36 parameters in a numerical experiment with D=5 1.008 1.000 1.000 1 p 33 1.007 1.000 1.000 1 p 34 1.173 1.000 1.000 1 p 35 1.009 1.000 1.000 1 p 36 1.001 1.000 1.000 1 Table 5. The evolution of the parameters as the number of time points increase, from 1 to 5.…”

Optimal Stefan problem

“…Therefore, it can't be treated as a Neumann condition, even if we include the free boundary as one of the control components. In [6] a new approach was developed based on the weak formulation of the multiphase Stefan problem as a boundary value problem for the nonlinear PDE with discontinuous coefficient. The optimal control framework was applied to the inverse multiphase Stefan problem with non-homogeneous Neumann conditions on the fixed boundaries in the case when the space dimension is one.…”