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doi: 10.3934/mcrf.2022008
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A differential game control problem with state constraints

1. 

ENSTA Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

2. 

Normandie University, INSA Rouen, Laboratoire Mathématiques INSA (LMI), 76000 Rouen, France

Received  June 2020 Revised  January 2022 Early access March 2022

Fund Project: 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.

Citation: Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022008
References:
[1]

A. AltaroviciO. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for non-linear state-constrained control problems, ESAIM Control Optim. Calc. Var., 19 (2013), 337-357.  doi: 10.1051/cocv/2012011.

[2]

M. AssellaouO. BokanowskiA. Desilles and H. Zidani, Value function and optimal trajectories for a maximum running cost control problem with state constraints. application to an abort landing problem, ESAIM Math. Model. Numer. Anal., 52 (2018), 305-335.  doi: 10.1051/m2an/2017064.

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[4]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[5]

M. BardiS. Koike and P. Soravia, Pursuit-evasion games with state constraints: Dynamic programming and discrete-time approximations, Discrete Contin. Dynam. Systems, 6 (2000), 361-380.  doi: 10.3934/dcds.2000.6.361.

[6]

P. BettiolP. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Internat. J. Game Theory, 34 (2006), 495-527.  doi: 10.1007/s00182-006-0030-9.

[7]

P. BettiolM. Quincampoix and R. Vinter, Existence and characterization of the values of two player differential games with state constraints, Appl. Math. Optim., 80 (2019), 765-799.  doi: 10.1007/s00245-019-09608-8.

[8]

O. BokanowskiN. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316.  doi: 10.1137/090762075.

[9]

N. Botkin and V. Turova, Dynamic programming approach to aircraft control in a windshear, Advances in Dynamic Games, 13 (2013), 53-69.  doi: 10.1007/978-3-319-02690-9_3.

[10]

R. BulirschF. Montrone and H. J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem, part 1: Necessary conditions, J. Optim. Theory Appl., 70 (1991), 1-23.  doi: 10.1007/BF00940502.

[11]

R. BulirschF. Montrone and H. J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem, part 2: Multiple shooting and homotopy, J. Optim. Theory Appl., 70 (1991), 223-254.  doi: 10.1007/BF00940625.

[12]

P. Cardaliaguet, A differential game with two players and one target: The continuous case, 1994.

[13]

P. Cardaliaguet, A differential game with two players and one target, SIAM J. Control Optim., 34 (1996), 1441-1460.  doi: 10.1137/S036301299427223X.

[14]

P. Cardaliaguet, Nonsmooth semipermeable barriers, Isaacs' equation, and application to a differential game with one target and two players, Appl. Math. Optim., 36 (1997), 125-146. 

[15]

P. CardaliaguetM. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints, SIAM J. Control Optim., 39 (2000), 1615-1632.  doi: 10.1137/S0363012998349327.

[16]

R. Elliott and N. Kalton, The Existence of Value in Differential Games, Memoirs of the American Mathematical Society, No. 126. American Mathematical Society, Providence, R.I., 1972.

[17]

R. Elliott and N. Kalton, Values in differential games, Bull. Amer. Math. Soc., 78 (1972), 427-431.  doi: 10.1090/S0002-9904-1972-12929-X.

[18]

H. FrankowskaS. Plaskacz and T. Rzezuchowski, Measurable viability theorems and the Hamilton-Jacobi-Bellman equation, J. Differential Equations, 116 (1995), 265-305.  doi: 10.1006/jdeq.1995.1036.

[19]

C. HermosillaP. R. Wolenski and H. Zidani, The mayer and minimum time problems with stratified state constraints, Set-Valued Var. Anal., 26 (2018), 643-662.  doi: 10.1007/s11228-017-0413-z.

[20]

A. MieleT. Wang and W. Melvin, Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance, J. Optim. Theory Appl., 58 (1988), 165-207.  doi: 10.1007/BF00939681.

[21]

A. MieleT. WangC. Tzeng and W. Melvin, Optimal abort landing trajectories in the presence of windshear, Journal of Optimization Theory and Applications, 55 (1987), 165-202. 

[22]

S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922.  doi: 10.1137/0728049.

[23]

M. Quincampoix and O. S. Serea, A viability approach for optimal control with infimum cost, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat., 48 (2002), 113-132. 

[24]

H. Rahimian and S. Mehrotra, Distributionally robust optimization: A review, Optimization Online, 2019.

[25]

E. A. Rapaport, Characterization of barriers of differential games, J. Optim. Theory Appl., 97 (1998), 151-179.  doi: 10.1023/A:1022631318424.

[26]

J. Rowland and R. Vinter, Construction of optimal feedback controls, Systems Control Lett., 16 (1991), 357-367.  doi: 10.1016/0167-6911(91)90057-L.

[27]

O. S. Serea, Discontinuous differential games and control systems with supremum cost, J. Math. Anal. Appl., 270 (2002), 519-542.  doi: 10.1016/S0022-247X(02)00087-2.

[28]

H. M. Soner, Optimal control with state-space constraint i, SIAM J. Control Optim., 24 (1986), 552-561.  doi: 10.1137/0324032.

show all references

References:
[1]

A. AltaroviciO. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for non-linear state-constrained control problems, ESAIM Control Optim. Calc. Var., 19 (2013), 337-357.  doi: 10.1051/cocv/2012011.

[2]

M. AssellaouO. BokanowskiA. Desilles and H. Zidani, Value function and optimal trajectories for a maximum running cost control problem with state constraints. application to an abort landing problem, ESAIM Math. Model. Numer. Anal., 52 (2018), 305-335.  doi: 10.1051/m2an/2017064.

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[4]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.

[5]

M. BardiS. Koike and P. Soravia, Pursuit-evasion games with state constraints: Dynamic programming and discrete-time approximations, Discrete Contin. Dynam. Systems, 6 (2000), 361-380.  doi: 10.3934/dcds.2000.6.361.

[6]

P. BettiolP. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Internat. J. Game Theory, 34 (2006), 495-527.  doi: 10.1007/s00182-006-0030-9.

[7]

P. BettiolM. Quincampoix and R. Vinter, Existence and characterization of the values of two player differential games with state constraints, Appl. Math. Optim., 80 (2019), 765-799.  doi: 10.1007/s00245-019-09608-8.

[8]

O. BokanowskiN. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316.  doi: 10.1137/090762075.

[9]

N. Botkin and V. Turova, Dynamic programming approach to aircraft control in a windshear, Advances in Dynamic Games, 13 (2013), 53-69.  doi: 10.1007/978-3-319-02690-9_3.

[10]

R. BulirschF. Montrone and H. J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem, part 1: Necessary conditions, J. Optim. Theory Appl., 70 (1991), 1-23.  doi: 10.1007/BF00940502.

[11]

R. BulirschF. Montrone and H. J. Pesch, Abort landing in the presence of windshear as a minimax optimal control problem, part 2: Multiple shooting and homotopy, J. Optim. Theory Appl., 70 (1991), 223-254.  doi: 10.1007/BF00940625.

[12]

P. Cardaliaguet, A differential game with two players and one target: The continuous case, 1994.

[13]

P. Cardaliaguet, A differential game with two players and one target, SIAM J. Control Optim., 34 (1996), 1441-1460.  doi: 10.1137/S036301299427223X.

[14]

P. Cardaliaguet, Nonsmooth semipermeable barriers, Isaacs' equation, and application to a differential game with one target and two players, Appl. Math. Optim., 36 (1997), 125-146. 

[15]

P. CardaliaguetM. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints, SIAM J. Control Optim., 39 (2000), 1615-1632.  doi: 10.1137/S0363012998349327.

[16]

R. Elliott and N. Kalton, The Existence of Value in Differential Games, Memoirs of the American Mathematical Society, No. 126. American Mathematical Society, Providence, R.I., 1972.

[17]

R. Elliott and N. Kalton, Values in differential games, Bull. Amer. Math. Soc., 78 (1972), 427-431.  doi: 10.1090/S0002-9904-1972-12929-X.

[18]

H. FrankowskaS. Plaskacz and T. Rzezuchowski, Measurable viability theorems and the Hamilton-Jacobi-Bellman equation, J. Differential Equations, 116 (1995), 265-305.  doi: 10.1006/jdeq.1995.1036.

[19]

C. HermosillaP. R. Wolenski and H. Zidani, The mayer and minimum time problems with stratified state constraints, Set-Valued Var. Anal., 26 (2018), 643-662.  doi: 10.1007/s11228-017-0413-z.

[20]

A. MieleT. Wang and W. Melvin, Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance, J. Optim. Theory Appl., 58 (1988), 165-207.  doi: 10.1007/BF00939681.

[21]

A. MieleT. WangC. Tzeng and W. Melvin, Optimal abort landing trajectories in the presence of windshear, Journal of Optimization Theory and Applications, 55 (1987), 165-202. 

[22]

S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907-922.  doi: 10.1137/0728049.

[23]

M. Quincampoix and O. S. Serea, A viability approach for optimal control with infimum cost, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat., 48 (2002), 113-132. 

[24]

H. Rahimian and S. Mehrotra, Distributionally robust optimization: A review, Optimization Online, 2019.

[25]

E. A. Rapaport, Characterization of barriers of differential games, J. Optim. Theory Appl., 97 (1998), 151-179.  doi: 10.1023/A:1022631318424.

[26]

J. Rowland and R. Vinter, Construction of optimal feedback controls, Systems Control Lett., 16 (1991), 357-367.  doi: 10.1016/0167-6911(91)90057-L.

[27]

O. S. Serea, Discontinuous differential games and control systems with supremum cost, J. Math. Anal. Appl., 270 (2002), 519-542.  doi: 10.1016/S0022-247X(02)00087-2.

[28]

H. M. Soner, Optimal control with state-space constraint i, SIAM J. Control Optim., 24 (1986), 552-561.  doi: 10.1137/0324032.

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
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
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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