Performance of a Markovian neural network versus dynamic programming on a fishing control problem

Figure 1.  Solution $ u(X,t) $ from SDP

Figure 2.  Values of the average cost versus $ X_0 $ for $ 100 $ realizations of $ {\rm {d}} W $ and with the SDP, for the 3 meshes, showing that the results are independent of $ n $

Figure 3.  Simulation of the fishing model with a quota function computed by SDP and $ X_0=0.7 $

Figure 4.  Simulation of the fishing model with a quota function computed by SDP and $ X_0=1.3 $

Figure 5.  SDP solution: $ V(X,t) $

Figure 6.  HJB solution: value function $ V(X,t) $

Figure 7.  Solution $ u(X,t) $ from HJB

Figure 8.  Values of the cost functions for the 3 space$ \times $time meshes $ 40n\times50n $, to solve the control problem by a HJB algorithm

Figure 9.  Simulation of the fishing model with a quota function computed by HJB and $ X_0=0.7 $

Figure 10.  Simulation of the fishing model with a quota function computed by HJB and $ X_0=1.3 $

Figure 11.  HJB: $ X_1,t\mapsto u_1 $

Figure 12.  HJB:$ X_2,t\mapsto u_1 $

Figure 13.  HJB:$ X_3,t\mapsto u_1 $

Figure 14.  HJB:$ X_1,t\mapsto u_2 $

Figure 15.  HJB:$ X_2,t\mapsto u_2 $

Figure 16.  HJB:$ X_3,t\mapsto u_2 $

Figure 17.  HJB:$ X_1,t\mapsto u_3 $

Figure 18.  HJB:$ X_2,t\mapsto u_3 $

Figure 19.  HJB:$ X_3,t\mapsto u_3 $

Figure 20.  HJB: Optimal biomass and quota function when X₁(0) = 0.7 and X₁(0) = 1.3

Figure 21.  HJB: Optimal biomass and quota function when X₂(0) = 0.7 and X₂(0) = 1.3

Figure 22.  HJB: Optimal biomass and quota function when X₃(0) = 0.7 and X₃(0) = 1.3

Figure 23.  Dynamic feedback control, $ u_\theta(X,t) $ computed by the Markovian Neural Network

Figure 24.  Single species: Values of the cost versus $ X_0 $ for 100 realizations with the Markovian neural network, for the 3 time meshes 50×n

Figure 25.  Simulation for a single fish species computed by the Markovian Neural Network and $ X_0=0.7 $. Performance with and without quota u_t

Figure 26.  Simulation for a single fish species computed by the Markovian Neural Network and $ X_0=1.3 $. Performance with and without quota $ u_t $

Figure 27.  $ X_1,t\mapsto u_1 $

Figure 28.  $ X_2,t\mapsto u_1 $

Figure 29.  $ X_3,t\mapsto u_1 $

Figure 30.  $ X_1,t\mapsto u_2 $

Figure 31.  $ X_2,t\mapsto u_2 $

Figure 32.  $ X_3,t\mapsto u_2 $

Figure 33.  $ X_1,t\mapsto u_3 $

Figure 34.  $ X_2,t\mapsto u_3 $

Figure 35.  $ X_3,t\mapsto u_3 $

Figure 36.  Optimal biomass and quota function computed with the two layers NN when X₀=0.7 and X₀=1.3

Figure 37.  Optimal biomass and quota function computed with the two layers NN when X₀=0.7 and X₀=1.3

Figure 38.  Optimal biomass and quota function computed with the two layers NN when $ X_0=0.7 $ and X₀ = 1.3

Figure 39.  Values of the average cost versus $ X_0 $ for $ 100 $ realizations of $ {\rm {d}} W $ and with the Markovian neural network with two layers of neurons

Figure 40.  5 species: $ X_1,t\mapsto u_1 $

Figure 41.  5 species:$ X_2,t\mapsto u_1 $

Figure 42.  5 species:$ X_5,t\mapsto u_1 $

Figure 43.  5 species:$ X_1,t\mapsto u_2 $

Figure 44.  5 species:$ X_2,t\mapsto u_2 $

Figure 45.  5 species:$ X_5,t\mapsto u_2 $

Figure 46.  5 species:$ X_1,t\mapsto u_3 $

Figure 47.  5 species:$ X_2,t\mapsto u_3 $

Figure 48.  5 species:$ X_5,t\mapsto u_3 $

Figure 49.  5 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $

Figure 50.  5 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $

Figure 51.  5 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $

Figure 52.  3 species:$ X_1,t\mapsto u_1 $

Figure 53.  3 species:$ X_2,t\mapsto u_1 $

Figure 54.  3 species:$ X_3,t\mapsto u_1 $

Figure 55.  3 species:$ X_1,t\mapsto u_2 $

Figure 56.  3 species:$ X_2,t\mapsto u_2 $

Figure 57.  3 species:$ X_3,t\mapsto u_2 $

Figure 58.  3 species:$ X_1,t\mapsto u_3 $

Figure 59.  3 species:$ X_2,t\mapsto u_3 $

Figure 60.  3 species:$ X_3,t\mapsto u_3 $

Figure 61.  3 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $

Figure 62.  3 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $

Figure 63.  3 species: Optimal biomass and quota function computed with the single layer NN when $ X_0=0.7 $ and $ X_0=1.3 $