Revert "Delete double pendulum"

This reverts commit 6f36279.

docs/src/examples/solvers/Double_Pendulum_Safety.jl

@@ -0,0 +1,54 @@
+using Test     #src
+using StaticArrays, Plots
+
+# At this point, we import the useful Dionysos sub-modules.
+using Dionysos
+const DI = Dionysos
+const UT = DI.Utils
+const DO = DI.Domain
+const ST = DI.System
+const SY = DI.Symbolic
+const OP = DI.Optim
+const AB = OP.Abstraction
+
+include(
+    joinpath(
+        dirname(dirname(pathof(Dionysos))),
+        "problems/double_pendulum",
+        "safety_stable_equilibrium.jl",
+    ),
+)
+
+# and we can instantiate the DC system with the provided system
+concrete_problem = DoublePendulum.problem(; approx_mode = "growth")
+concrete_system = concrete_problem.system
+
+x0 = SVector(0.0, 0.0, 0.0, 0.0)
+hx = SVector(0.1, 0.1, 0.1, 0.1)
+state_grid = DO.GridFree(x0, hx)
+
+u0 = SVector(0.0);
+h = SVector(0.3);
+input_grid = DO.GridFree(u0, h);
+
+using JuMP
+optimizer = MOI.instantiate(AB.UniformGridAbstraction.Optimizer)
+MOI.set(optimizer, MOI.RawOptimizerAttribute("concrete_problem"), concrete_problem)
+MOI.set(optimizer, MOI.RawOptimizerAttribute("state_grid"), state_grid)
+MOI.set(optimizer, MOI.RawOptimizerAttribute("input_grid"), input_grid)
+MOI.optimize!(optimizer)
+
+abstract_controller = MOI.get(optimizer, MOI.RawOptimizerAttribute("abstract_controller"))
+concrete_controller = MOI.get(optimizer, MOI.RawOptimizerAttribute("concrete_controller"))
+
+# ### Trajectory display
+# We choose the number of steps `nsteps` for the sampled system, i.e. the total elapsed time: `nstep`*`tstep`
+# as well as the true initial state `x0` which is contained in the initial state-space defined previously.
+nstep = 100
+x0 = SVector(0.15, 0.0, 0.0, 0.0) # SVector(0.15,0.0) #
+control_trajectory =
+    ST.get_closed_loop_trajectory(concrete_system.f, concrete_controller, x0, nstep)
+
+fig = plot(; aspect_ratio = :equal);
+plot!(concrete_system.X);
+plot!(control_trajectory; markersize = 1, arrows = false)