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Abstract Givenndisjoint intervals on together withnfunctions , , and an matrix , the problem is to find anL2solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem onncopies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.
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This content will become publicly available on January 1, 2026
Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations
In this article, we provide an original systematic global-in-time analysis of mean field type control problems on ℝnwith generic cost functions allowing quadratic growth by a novel “lifting” approach which is not the same as the traditional lifting. As an alternative to the recent popular analytical method of tackling master equations, we resolve the control problem in a proper Hilbert subspace of the whole space ofL2random variables, it can be regarded as a tangent space attached at the initial probability measure. The problem is linked to the global solvability of the Hilbert-space-valued forward–backward stochastic differential equation (FBSDE), which is solved by variational techniques here. We also rely on the Jacobian flow of the solution to this FBSDE to establish the regularity of the value function, including its linearly functional differentiability, which leads to the classical wellposedness of the Bellman equation. Together with the linear functional derivatives and the gradient of the linear functional derivatives of the solution to the FBSDE, we also obtain the classical wellposedness of the master equation. Our current approach imposes structural conditions directly on the cost functions. The contributions of adopting this framework in our study are twofold: (i) compared with imposing conditions on Hamiltonian, the structural conditions imposed in this work are easily verified, and less demanding on the cost functions while solving the master equation; and (ii) when the cost functions are not convex in the state variable or there is a lack of monotonicity of cost functions, an accurate lifespan can be provided for the local existence, which may not be that small in many cases.
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- Award ID(s):
- 2204795
- PAR ID:
- 10627993
- Publisher / Repository:
- EDP Sciences
- Date Published:
- Journal Name:
- ESAIM: Control, Optimisation and Calculus of Variations
- Volume:
- 31
- ISSN:
- 1292-8119
- Page Range / eLocation ID:
- 33
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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