Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions
Tóm tắt
The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ
N
evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.
Tài liệu tham khảo
Aubin, J.-P.: Mutational and Morphological Analysis: Tools for Shape Evolution and Morphogenesis. Birkhäuser, Boston (1999)
Aubin, J.-P.: Mutational equations in metric spaces. Set-Valued Anal. 1, 3–46 (1993)
Aubin, J.-P.: A note on differential calculus in metric spaces and its applications to the evolution of tubes. Bull. Polish Acad. Sci. Math. 40(2), 151–162 (1992)
Aubin, J.-P.: Viability Theory. Birkhäuser, Boston (1991)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Aubin, J.-P., Murillo Hernández, J.A.: Morphological equations and sweeping processes. In: Alart, P., et al. (eds.) Nonsmooth Mechanics and Analysis. Theoretical and Numerical Advances, pp. 249–259. Springer, Berlin Heidelberg New York (2006)
Cannarsa, P., Frankowska, H.: Interior sphere property of attainable sets and time optimal control problems. ESAIM Control Optim. Calc. Var. 12, 350–370 (2006)
Cardaliaguet, P.: Front propagation problems with nonlocal terms II. J. Math. Anal. Appl. 260(2), 572–601 (2001)
Cardaliaguet, P.: On front propagation problems with nonlocal terms. Adv. Differential Equations 5(1–3), 213–268 (2000)
Caroff, N., Frankowska, H.: Conjugate points and shocks in nonlinear optimal control. Trans. Amer. Math. Soc. 348(8), 3133–3153 (1996)
Cornet, B., Czarnecki, M.-O.: Smooth normal approximations of epi-Lipschitzian subsets of ℝn, SIAM J. Control Optim. 37(3), 710–730 (1999)
Delfour, M.C., Zolésio, J.-P.: Shapes and geometries. Analysis, differential calculus, and optimization. SIAM Adv. Des. Control 4, (2001)
Doyen, L.: Mutational equations for shapes and vision-based control. J. Math. Imaging Vision 5(2), 99–109 (1995)
Doyen, L.: Filippov and invariance theorems for mutational inclusions of tubes. Set-Valued Anal. 1(3), 289–303 (1993)
Doyen, L., Najman, L., Mattioli, J.: Mutational equations of the morphological dilation tubes. J. Math. Imaging Vision 5(3), 219–230 (1995)
Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)
Frankowska, H.: Value function in optimal control. In: Agrachev, A.A. (ed.) Mathematical control theory. ICTP Lecture Notes, vol. 8, pp. 515–653. ICTP, Trieste (2002)
Gorre, A.: Evolutions of tubes under operability constraints. J. Math. Anal. Appl. 216(1), 1–22 (1997)
Lorenz, T.: Radon measures solving the Cauchy problem of the nonlinear transport equation. (2007) IWR Preprint at http://www.ub.uni-heidelberg.de/archiv/7252
Lorenz, T.: Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms. Submitted to Discuss. Math. Differential Incl. Control Optim. (2006) (http://www.ub.uni-heidelberg.de/archiv/7392)
Lorenz, T.: Boundary regularity of reachable sets of control systems. Systems Control Lett. 54(9), 919–924 (2005)
Lorenz, T.: Evolution equations in ostensible metric spaces: definitions and existence. (2005) IWR Preprint at http://www.ub.uni-heidelberg.de/archiv/5519
Lorenz, T.: First-order geometric evolutions and semilinear evolution equations: a common mutational approach. Doctor thesis, Ruprecht-Karls University of Heidelberg (2004) http://www.ub.uni-heidelberg.de/archiv/4949
Najman, L.: Euler method for mutational equations. J. Math. Anal. Appl. 196(3), 814–822 (1995)
Panasyuk, A.I.: Quasidifferential equations in a complete metric space under conditions of the Carathéodory type. I. Differential Equations 31(6), 901–910 (1995)
Panasyuk, A.I.: Quasidifferential equations in a complete metric space under Carathéodory-type conditions. II. Differential Equations 31(8), 1308–1317 (1995)
Panasyuk, A.I.: Quasidifferential equations in metric spaces. Differential Equations 21, 914–921 (1985)
Reilly, I.L., Subrahmanyam, P.V., Vamanamurthy, M.K.: Cauchy sequences in quasi-pseudo-metric spaces. Monatsh. Math. 93, 127–140 (1982)
Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Grundlehren der mathematischen Wissenschaften 317. Springer, Berlin Heidelberg New York (1998)
Royden, H.L.: Comparison theorems for the matrix Riccati equation. Comm. Pure Appl. Math. 41(5), 739–746 (1988)
Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)