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# A differential game control problem with state constraints

The second author acknowledges a support from the Normandy Region and the EU through ERDF program under grant "Chaire d'excellence COPTI"

• We study the Hamilton-Jacobi (HJ) approach for a two-person zero-sum differential game with state constraints and where controls of the two players are coupled within the dynamics, the state constraints and the cost functions. It is known for such problems that the value function may be discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. In this work, we characterize this value function through an auxiliary differential game free of state constraints. Furthermore, we establish a link between the optimal strategies of the constrained problem and those of the auxiliary problem and we present a general approach allowing to construct approximated optimal feedbacks to the constrained differential game for both players. Finally, an aircraft landing problem in the presence of wind disturbances is given as an illustrative numerical example.

Mathematics Subject Classification: Primary: 49N70, 49L20; Secondary: 49N90.

 Citation:

• Figure 1.  Trajectories and controls reconstruction as a response to arbitrary disturbances (random distribution, Algorithm 1, in blue) and to the worst case (Algorithm 2, in red) from $y_1$

Figure 2.  Trajectories and controls reconstruction as a response to arbitrary disturbances (random distribution, Algorithm 1, in blue) and to the worst case (Algorithm 2, in red) from $y_2$

Figure 3.  Trajectories and controls reconstruction as a response to arbitrary disturbances (random distribution, Algorithm 1, in blue) and to the worst case (Algorithm 2, in red) from $y_3$

Table 1.  State constraints: domain $\mathcal{K}$

 State variable $h$(ft) $v$(ft $s^{-1}$) $\gamma$ (deg) $w_h$(ft $s^{-1}$) $\alpha$ (deg) min 450 160 -7.0 -100.0 0.0 max 1000 260 15.0 50.0 17.2

Table 2.  Control constraints: sets $A$ and $B$

 Control variables $a$ (deg $s^{-1}$) $b_1:=\dot{w_x}$(ft $s^{-2}$) $b_2:=\dot{w_h}$(ft $s^{-2}$) min -3.0 0.0 -2.0 max 3.0 7.7 2.0
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