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Constrained Evolution for a Quasilinear Parabolic Equation

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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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References

  1. Pisano, A., Usai, E.: Sliding mode control: a survey with applications in math. Math. Comput. Simul. 81, 954–979 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Levaggi, L.: Infinite dimensional systems’ sliding motions. Eur. J. Control 8, 508–516 (2002)

    Article  MATH  Google Scholar 

  3. Cheng, M.-B., Radisavljevic, V., Su, W.-C.: Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Autom. J. IFAC 47, 381–387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Levaggi, L.: Existence of sliding motions for nonlinear evolution equations in Banach spaces. Discrete Contin. Dyn. Syst. In: 9th AIMS Conference on Dynamical Systems, Differential Equations and Applications, pp. 477–487 (2013)

  5. Xing, H., Li, D., Gao, C., Kao, Y.: Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay. J. Franklin Inst. 350, 397–418 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbu, V., Colli, P., Gilardi, G., Marinoschi, G., Rocca, E.: Sliding mode control for a nonlinear phase-field system. Preprint. arXiv:1506.01665 [math.AP], pp. 1-28 (2015)

  7. Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)

    Book  Google Scholar 

  8. Brezis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud. 5. North-Holland, Amsterdam (1973)

  9. DiBenedetto, E.: Degenerate Parabolic Equations, Universitext. Springer, New York (1993)

    Book  MATH  Google Scholar 

  10. Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod; Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  11. Roubíček, T.: Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153. Birkhäuser Verlag, Basel (2005)

  12. Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs 49. American Mathematical Society, Providence (1997)

    Google Scholar 

  13. Zheng, S.: Nonlinear Evolution Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 133. Chapman & Hall/CRC, Boca Raton (2004)

    Google Scholar 

  14. Fitzpatrick, S., Phelps, R.R.: Differentiability of the metric projection in Hilbert space. Trans. Am. Math. Soc. 270, 483–501 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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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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Correspondence to Pierluigi Colli.

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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

$$\begin{aligned} Dd_K(v) = \frac{Q_Kv}{\mathopen \Vert Q_Kv\mathclose \Vert _H}. \end{aligned}$$
(79)

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:

$$\begin{aligned} Dd_K(v) = \frac{1}{2} \, (\varphi (v))^{-1/2} D\varphi (v) = \frac{1}{2} (d_K(v))^{-1} \, 2Q_Kv = \frac{Q_Kv}{\mathopen \Vert Q_Kv\mathclose \Vert }. \end{aligned}$$

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

$$\begin{aligned} \xi := \frac{Q_K(v)}{d_K(v)} \end{aligned}$$

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

$$\begin{aligned} v - \varepsilon \xi - P_{K}v = Q_Kv - \frac{\varepsilon }{d_K(v)} \, Q_Kv = \lambda Q_Kv = \lambda (v-P_{K}v) . \end{aligned}$$
(80)

Then, we have, for every \(z\in K\),

$$\begin{aligned} (v-\varepsilon \xi -P_{K}v,z-P_{K}v) = \lambda (v-P_{K}v,z-P_{K}v) \le 0. \end{aligned}$$

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

$$\begin{aligned} Dd_K(v-\varepsilon \xi ) = \frac{v-\varepsilon \xi -P_{K}v}{\mathopen \Vert v-\varepsilon \xi -P_{K}v\mathclose \Vert } = \frac{\lambda Q_Kv}{\mathopen \Vert \lambda Q_Kv\mathclose \Vert } = \xi . \end{aligned}$$

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

$$\begin{aligned} d^{\,\varepsilon }_K(v) = \frac{1}{2\varepsilon } \, {(d_K(v))^2} \quad \hbox {if }0<d_K(v)<\varepsilon . \end{aligned}$$
(81)

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

$$\begin{aligned} \frac{z-v}{\varepsilon }+ \frac{Q_Kz}{\mathopen \Vert Q_Kz\mathclose \Vert } = 0 , \quad \hbox {that is} , \quad v = z + \varepsilon \, \frac{Q_Kz}{\mathopen \Vert Q_Kz\mathclose \Vert }. \end{aligned}$$

It easily follows that \(P_{K}v=P_{K}z\). Therefore, we have that

$$\begin{aligned} d_K(v) = \mathopen \Vert v-P_{K}v\mathclose \Vert = \mathopen \Vert v-P_{K}z\mathclose \Vert = \Bigl ( 1 + \frac{\varepsilon }{\mathopen \Vert Q_Kz\mathclose \Vert } \Bigr ) \mathopen \Vert z-P_{K}z\mathclose \Vert = d_K(z) + \varepsilon > \varepsilon , \end{aligned}$$

while we were assuming that \(d_K(v)<\varepsilon \). Therefore, every minimum point z has to belong to K, and we have that

$$\begin{aligned} d^{\,\varepsilon }_K(v) = \min _{z\in K} \frac{1}{2\varepsilon } \, \mathopen \Vert z-v\mathclose \Vert ^2 = \frac{1}{2\varepsilon } \, \mathopen \Vert v-P_{K}v\mathclose \Vert ^2 , \end{aligned}$$

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

$$\begin{aligned} \min \bigl \{ d_K(v)/\varepsilon , 1 \bigr \} Dd_K(v) = \min \bigl \{ d_K(v)/\varepsilon , 1 \bigr \} \, \frac{Q_Kv}{d_K(v)} = \frac{Q_Kv}{\max \{\varepsilon ,d_K(v)\}} = Dd^{\,\varepsilon }_K(v) . \end{aligned}$$

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

$$\begin{aligned} 0 \le \int _0^{d_K(v+h)} \min \{s/\varepsilon ,1\} \, \mathrm{d}s = \int _0^{d_K(v+h)} \frac{s}{\varepsilon }\, \mathrm{d}s = \frac{1}{2\varepsilon } \, {(d_K(v+h))^2} \le \frac{1}{2\varepsilon } \, \mathopen \Vert h\mathclose \Vert ^2. \end{aligned}$$

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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