Abstract
In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy–Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set of the space of square-integrable functions. Then, we consider convex sets of obstacle or double-obstacle type and prove rigorously the following property: if the factor in front of the feedback control is sufficiently large, then the solution reaches the convex set within a finite time and then moves inside it.
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Acknowledgments
PC and GG gratefully acknowledge some financial support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia.
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Communicated by Irena Lasiecka.
Appendix
Appendix
In this section, we prove Lemma 5.1. We denote by \((\,\cdot \,,\,\cdot \,)\) the scalar product of H and write \(\mathopen \Vert \,\cdot \,\mathclose \Vert \) instead of \(\mathopen \Vert \,\cdot \,\mathclose \Vert _H\). We start proving that the function \(d_K\) is Fréchet differentiable at any point \(v\in H\setminus K\) and that its gradient is given by
This easily follows from [14, Prop. 2.2]. Indeed, we immediately deduce from this result that \(2Q_K\) is the Fréchet derivative of the map \(v\mapsto \varphi (v):=\mathopen \Vert Q_Kv\mathclose \Vert ^2\). Now, we read \(d_K\) as the square root of \(\varphi \) and assume that \(v\in H\setminus K\), i.e., \(\varphi (v)>0\). Then, \(\varphi >0\) in a neighborhood of v, whence \(d_K\) is Fréchet differentiable at v and (79) follows from applying the chain rule:
On the contrary, the proof of the rest of the lemma needs some work. Assume first \(v\in K\). Then, v is a minimum point for \(d_K\) and \(d^{\,\varepsilon }_K\), whence \(0\in \partial d_K(v)\) and \(Dd^{\,\varepsilon }_K(v)=0\). On the other hand, we have \(Q_Kv=0\) , and thus, (37) holds true in this case. Assume now \(d_K(v)>\varepsilon \). Then, the point
satisfies \(\mathopen \Vert \xi \mathclose \Vert =1\) and \(v-\varepsilon \xi \not \in K\). Hence, (37) reduces to \(Dd^{\,\varepsilon }_K(v)=\xi \) and thus means that \(Dd_K(v-\varepsilon \xi )=\xi \) (by the definition of the Yosida regularization). Therefore, we prove this fact. We set \(\lambda :=1-\varepsilon /d_K(v)\) and observe that \(\lambda >0\) and that
Then, we have, for every \(z\in K\),
As \(P_{K}v\in K\), this shows that \(P_{K}(v-\varepsilon \xi )=P_{K}(v)\). By applying (79) to \(v-\varepsilon \xi \) and recalling (80) once more, we obtain
Hence, the desired equality is established under the assumption \(d_K(v)>\varepsilon \). As \(d^{\,\varepsilon }_K\), \(Q_K\) and \(d_K\) are continuous, (37) holds also if \(d_K(v)=\varepsilon \), and we consider the last case, i.e., \(0<d_K(v)<\varepsilon \). We first prove that
It is well known that the infimum in the definition (35) is a minimum. We first look for a minimum point \(z\not \in K\). Then, in view of (79), z has to satisfy
It easily follows that \(P_{K}v=P_{K}z\). Therefore, we have that
while we were assuming that \(d_K(v)<\varepsilon \). Therefore, every minimum point z has to belong to K, and we have that
so that (81) is proved. Since the set \(\mathopen \{v\in H:\ 0<d_K(v)<\varepsilon \mathclose \}\) is open, we can differentiate (81) by applying the chain rule and (79) and deduce that (37) holds also in this case. To conclude the proof, we have to derive (38). Now observe that this identity trivially holds if \(v\in K\) and that K is nonempty. It thus suffices to prove that both sides of the identity have the same Fréchet gradient. To this end, assume first that \(v\not \in K\). By differentiating the right-hand side at v with the chain rule and applying (79) and (37), we obtain
Assume now that v belongs to K and take any \(h\in H\) satisfying \(\mathopen \Vert h\mathclose \Vert \le \varepsilon \). Then, we have \(d_K(v+h)\le \mathopen \Vert h\mathclose \Vert \le \varepsilon \), and we infer that
Thus, the Fréchet gradient of the right-hand side of (38) at v is zero. On the other hand, we also have \(Dd^{\,\varepsilon }_K(v)=0\) in this case. This completes the proof. \(\square \)
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Colli, P., Gilardi, G. & Sprekels, J. Constrained Evolution for a Quasilinear Parabolic Equation. J Optim Theory Appl 170, 713–734 (2016). https://doi.org/10.1007/s10957-016-0970-6
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DOI: https://doi.org/10.1007/s10957-016-0970-6