An existence result for optimal control of singular constraints involving 1-Laplacian operator

References

• Liu R, Lin Z, Zhang W, et al. Learning PDEs for image restoration via optimal control. In: European Conference on Computer Vision. Springer; 2010. p. 115–128.
   Google Scholar
• Liu R, Lin Z, Zhang W, et al. Toward designing intelligent PDEs for computer vision: an optimal control approach. Image Vision Comput. 2013;31(1):43–56.
   Web of Science ®Google Scholar
• Casas E. Optimal control of PDE theory and numerical analysis. Castro Urdiales: CIMPA School on Optimization and Control; 2006.
   Google Scholar
• Li X, Yong J. Optimal control theory for infinite dimensional systems. Boston (MA): Birkhauser; 1995.
   Google Scholar
• Lions JL. Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Heidelberg: Springer Verlag; 1968.
   Google Scholar
• Tröltzsch F. Optimal control of partial differential equations: theory, methods, and applications. Providence (RI): American Mathematical Soc; 2010.
   Google Scholar
• Borwein JM. Fifty years of maximal monotonicity. Optim Lett. 2010;4(4):473–490.
   Web of Science ®Google Scholar
• Rockafellar R. On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc. 1970;149(1):75–88.
   Web of Science ®Google Scholar
• Simons S, Zalinescu C. Fenchel duality, fitzpatrick functions and maximal monotonicity. J Nonlin Convex Anal. 2005;6(1):1–22.
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in hilbert spaces. 2nd ed. New York (NY): Springer; 2017.
   Google Scholar
• Barbu V. Nonlinear differential equations of monotone types in banach spaces. New York (NY): Springer Science & Business Media; 2010.
   Google Scholar
• Showalter RE. Monotone operators in banach space and nonlinear partial differential equations. Providence (RI): American Mathematical Soc; 2013.
   Google Scholar
• Andreu F, Mazón JM. Parabolic quasilinear equations minimizing linear growth functionals. Basel: Springer Science & Business Media; 2004.
   Google Scholar
• Andreu F, Ballester C, Caselles V, et al. The dirichlet problem for the total variation flow. J Funct Anal. 2001;180(2):347–403.
   Web of Science ®Google Scholar
• Ambrosio L, Fusco N, Pallara D. Functions of bounded variation and free discontinuity problems. Oxford: Clarendon Press; 2000.
   Google Scholar
• Giusti E, Williams GH. Minimal surfaces and functions of bounded variation. Boston (MA): Springer; 1984.
   Google Scholar
• Ziemer WP. Weakly differentiable functions: Sobolev spaces and functions of bounded variation. New York (NY): Springer Science & Business Media; 1989.
   Google Scholar
• Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Heidelberg: Springer; 2001.
   Google Scholar
• Ekeland I, Temam R. Convex analysis and variational problems. Philadelphia (PA): SIAM; 1999.
   Google Scholar
• Aubert G, Kornprobst P. Mathematical problems in image processing: partial differential equations and the calculus of variations. New York (NY): Springer Science & Business Media; 2006.
   Google Scholar
• Chan TF, Shen J. Image processing and analysis: variational, pde, wavelet, and stochastic methods. Philadelphia (PA): SIAM; 2005.
   Google Scholar
• Tikhonov AN, Arsenin VY. Solutions of ill-posed problems. Washington (DC): John Wiley & Sons, Inc.; 1977.
   Google Scholar
• Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D. 1992;60(1–4):259–268.
   Web of Science ®Google Scholar
• Zhao Z, Fang C, Lin Z, et al. A robust hybrid method for text detection in natural scenes by learning-based partial differential equations. Neurocomputing. 2015;168:23–34.
   Web of Science ®Google Scholar
• Zhao Z, Lin Z, Wu Y. A fast alternating time-splitting approach for learning partial differential equations. Neurocomputing. 2016;185:171–182.
   Web of Science ®Google Scholar
• Fang C, Zhao Z, Zhou P. Feature learning via partial differential equation with applications to face recognition. Pattern Recognit. 2017;69:14–25.
   Web of Science ®Google Scholar
• Bénilan P, Crandall M, Pazy A. Nonlinear evolution equations governed by accretive operators. preprint book. 2001.
   Google Scholar
• Devi A, Kumar A, Baleanu D. On stability analysis and existence of positive solutions for a general nonlinear fractional differential equations. Adv Differ Equ. 2020;2020:1–16.
   PubMedGoogle Scholar