Optimistic Value Model of Uncertain Optimal Control

Math Methods in App Sciences

2018

Self Cite

The saddle point equilibrium problem is of great importance in differential game theory. In this paper, an optimistic value model for the saddle point equilibrium problem under uncertain environment is investigated. The equilibrium equation for the proposed model is presented. Then a linear quadratic model is discussed. Finally, a counter terror problem is analyzed by the results obtained in the paper.

“…Theorem (Sheng and Zhu) Let V ( t , x ) be twice differentiable on

false[0, T false] \times R^{n}

. Then we have

- V_{t} (t, x) = \sup_{u \in U} \{F (x, u, t) + \nabla_{bold-italicx} V {(t, x)}^{τ} f (x, u, t) + \frac{\sqrt{3}}{π} \ln \frac{1 - α}{α} ‖ \nabla_{bold-italicx} V {(t, x)}^{τ} g (x, u, t) {false‖}_{1}\},

where V t ( t , x ) is the partial derivatives of the function V ( t , x ) in t, ∇ x V ( t , x ) is the gradient of V ( t , x ) in x and ‖·‖ 1 is the 1‐norm for vectors, ie,

false‖ bold-italicp ‖_{1} = true \sum_{i = 1}^{n} false| p_{i} false|

for p =( p 1 , p 2 ,⋯, p n ) τ .…”

Section: Preliminarymentioning

confidence: 99%

“…The main problem is

\{\begin{matrix} V false(t, bold-italicx false) \equiv \sup_{u_{s} \in double-struckU false[t, T false]} E ([], \int_{t}^{T} F false(X_{s}, u_{s}, s false) normald s + h false(X_{T}, T false)) \\ subject to \\ normald X_{s} = bold-italicf false(X_{s}, u_{s}, s false) normald s + bold-italicG false(X_{s}, u_{s}, s false) normald C_{s} and X_{t} = bold-italicx, \end{matrix}

which was investigated using expected value criterion. In 2013, Sheng and Zhu considered the following uncertain optimistic value optimal control model:

\{\begin{matrix} V false(t, bold-italicx false) \equiv \sup_{u_{s} \in double-struckU false[t, T false]} H_{\sup} false(α false) \\ subject to \\ normald X_{s} = bold-italicf false(X_{s}, u_{s}, s false) normald s + bold-italicg false(X_{s}, u_{s}, s false) normald C_{s} and X_{t} = bold-italicx, \end{matrix}

where

H = \int_{t}^{T} F false(X_{s}, u_{s}, s false) normald s + G false...

…”

Section: Introductionmentioning

confidence: 99%

Saddle point equilibrium under uncertain environment

Sun

Ding

Math Methods in App Sciences

2018

Self Cite

“…In 2013, Sheng and Zhu (2013) considered the following uncertain optimistic value optimal control model:…”

Section: Preliminarymentioning

confidence: 99%

Bang–bang property for an uncertain saddle point problem

Sun

2014

J Intell Manuf

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“…As a model for dealing with human uncertainty, this theory has been successfully applied to many fields, such as uncertain programming, 11 uncertain finance, 12 uncertain differential game, 13 and uncertain supply chain. 14 In order to handle an optimal control problem in the uncertain environment, Zhu 15 proposed an uncertain optimal control model in 2010 and gave an optimality equation as a counterpart of Hamilton-Jacobi-Bellman equation in stochastic optimal control, Sheng and Zhu 16 established an optimistic value model for uncertain optimal control and gave the equation of optimality by applying the Bellman's principle of optimality, Yan and Zhu 17 studied an Bang-bang control model with optimistic value criterion for uncertain switched systems and proposed a two-stage algorithm to handle such model, Li and Zhu 18 introduced a parametric optimal control problem of uncertain linear quadratic model and proposed an approximation method to solve it.…”

Section: Introductionmentioning

confidence: 99%

Indefinite LQ optimal control with cross term for discrete‐time uncertain systems

Chen

Math Methods in App Sciences

2018

Self Cite

Recently, there has been an increasing interest in the study on uncertain optimal control problems. In this paper, a linear quadratic (LQ) optimal control with cross term for discrete‐time uncertain systems is considered, whereas the weighting matrices in the cost function are allowed to be indefinite. Firstly, a recurrence equation for the problem is presented based on Bellman's principle of optimality in dynamic programming. Then, a necessary condition for the existence of an optimal linear state feedback control of the indefinite LQ problem is given by the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ problem is presented by introducing a linear matrix inequality (LMI) condition. Furthermore, it is shown that the well‐posedness of the indefinite LQ problem, the solvability of the indefinite LQ problem, the LMI condition, and the solvability of the constrained difference equation are equivalent to each other. Finally, an example is presented to illustrate the results obtained.