Abstract
In this chapter, we derive a general Pontryagin-type stochastic maximum principle for optimal controls with a possibly nonconvex control domain.
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Notes
- 1.
Recall that, for any \(C^2\)-function \(f(\cdot )\) defined on a Banach space \(X\) and \(x_0\in X\), \(f_{xx}(x_0)\in {\fancyscript{L}}(X\times X,X)\). This means that, for any \(x_1,x_2\in X\), \(f_{xx}(x_0)(x_1,x_2)\in X\). Hence, by (9.6), \(a_{11}(t)\big (x_2^\varepsilon ,x_2^\varepsilon \big )\) [in (9.12)] stands for \(a_{xx}(t,\bar{x}(t),\bar{u}(t))\big (x_2^\varepsilon (t),x_2^\varepsilon (t)\big )\). One has a similar meaning for \(b_{11}(t)\big (x_2^\varepsilon ,x_2^\varepsilon \big )\) and so on.
- 2.
We have dropped this technical condition in the paper “Lü Q., Zhang, X.: Transposition method for backward stochastic evolution equations revisited, and its application. Math. Control Relat. Fields, In submission (See also http://arxiv.org/abs/1405.4454v1)”.
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Lü, Q., Zhang, X. (2014). Necessary Condition for Optimal Controls, the Case of Non-convex Control Domains. In: General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06632-5_9
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DOI: https://doi.org/10.1007/978-3-319-06632-5_9
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