Optimal growth for linear processes with affine control
Optimal growth for linear processes with affine control
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发表时间：
2012-03
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通讯作者：
V. Calvez;Pierre Gabriel
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作者：
V. Calvez;Pierre Gabriel
我们分析具有以下特征的最优控制：动力系统是线性的，并且对控制参数的依赖性是ane。更准确地说，我们考虑 _ x (t) = (G+ (t)F )x (t)，其中 G 和 F 是具有某种规定结构的 3 3 矩阵。在恒定控制 (t) 的情况下，我们证明了在某些假设下相对于变化存在最优 Perron 特征值。接下来我们研究与时间周期控制 (t) 相关的 Floquet 特征值问题。最后我们证明最优控制问题的特征值（广义上）的存在性。该证明基于 [Arisawa 1998, Ann. Institut Henri Poincar e] 关于 Hamilton-Jacobi 方程的遍历问题。我们讨论三个特征值之间的关系。令人惊讶的是，这三个特征值在数值上似乎是相同的。
We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is ane. More precisely we consider _ x (t) = (G+ (t)F )x (t), where G and F are 3 3 matrices with some prescribed structure. In the case of constant control (t) , we show the existence of an optimal Perron eigenvalue with respect to varying under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls (t). Finally we prove the existence of an eigenvalue (in the generalized sense) for the optimal control problem. The proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar e] concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the relations between the three eigenvalues. Surprisingly enough, the three eigenvalues appear to be numerically the same.