diff --git a/dynSys/@MountainCarV0/MountainCarV0.m b/dynSys/@MountainCarV0/MountainCarV0.m
new file mode 100644
index 0000000..684eafc
--- /dev/null
+++ b/dynSys/@MountainCarV0/MountainCarV0.m
@@ -0,0 +1,33 @@
+classdef MountainCarV0 < DynSys
+  properties
+    gravity
+    force
+  end
+  
+  methods
+    function obj = MountainCarV0(x, gravity, force)
+      
+      if numel(x) ~= 2
+        error('Initial state does not have right dimension!');
+      end
+      
+      if ~iscolumn(x)
+        x = x';
+      end
+      
+      
+      obj.x = x;
+      obj.xhist = obj.x;
+      
+      obj.pdim = 1; % position dimensions
+      
+      obj.nx = 2; % number of state dimensions
+      obj.nu = 1; % number of control dimensions
+      
+      obj.gravity = gravity; % gravity (default is 0.0025)
+      obj.force = force; % force (default is 0.001)
+      
+    end
+    
+  end % end methods
+end % end classdef
diff --git a/dynSys/@MountainCarV0/dynamics.m b/dynSys/@MountainCarV0/dynamics.m
new file mode 100644
index 0000000..4a1dfd0
--- /dev/null
+++ b/dynSys/@MountainCarV0/dynamics.m
@@ -0,0 +1,30 @@
+function dx = dynamics(obj, t, x, u, d)
+
+    gravity = obj.gravity;
+    force = obj.force;
+    
+%         velocity += (action - 1) * self.force + math.cos(3 * position) * (-self.gravity)
+%         velocity = np.clip(velocity, -self.max_speed, self.max_speed)
+%         position += velocity
+%         position = np.clip(position, self.min_position, self.max_position)
+if iscell(x)
+  dx = cell(obj.nx, 1);
+  
+  position = x{1};
+  velocity = x{2};
+  
+  dx{1} = velocity;
+  dx{2} = (u - 1) * force + cos(3 * position) * (-gravity);
+else
+  dx = zeros(obj.nx, 1);
+
+  position = x(1);
+  velocity = x(2);
+   
+  dx(1) = velocity;
+  % u ~ { 0, 1, 2 }
+  dx(2) = (u - 1) * force + cos(3 * position) * (-gravity);
+end
+
+
+end
\ No newline at end of file
diff --git a/dynSys/@MountainCarV0/optCtrl.m b/dynSys/@MountainCarV0/optCtrl.m
new file mode 100644
index 0000000..b103a19
--- /dev/null
+++ b/dynSys/@MountainCarV0/optCtrl.m
@@ -0,0 +1,39 @@
+function uOpt = optCtrl(obj, t, xs, deriv, uMode, ~)
+% uOpt = optCtrl(obj, t, deriv, uMode, dMode, MIEdims)
+
+%% Input processing
+if nargin < 5
+  uMode = 'min';
+end
+
+if ~iscell(deriv)
+  deriv = num2cell(deriv);
+end
+
+
+%% Optimal control
+
+% https://arxiv.org/pdf/1709.07523.pdf
+% Appendix 
+% need dot product of deriv and dx = f(x)
+% then argmin terms with action to find optimal control
+
+% we have    
+%   dx(1) = velocity;
+% u ~ { 0, 1, 2 }
+%   dx(2) = (u - 1) * force + cos(3 * position) * (-gravity);
+
+% u* = argmin_u (deriv{[0, 1]} * dx(2))
+%    = argmin_u (deriv{[0, 1]} (u - 1) * force)
+%    = 0 when deriv{[0, 1]} >= 0, 2 when deriv{[0, 1]} < 0
+
+%% Optimal control
+if strcmp(uMode, 'max')
+  uOpt = (deriv{1} >= 0) * 2 + (deriv{1} < 0) * 0;
+elseif strcmp(uMode, 'min')
+  uOpt = (deriv{1} >= 0) * 0 + (deriv{1} < 0) * 2;
+else
+  error('Unknown uMode!')
+end
+
+end
\ No newline at end of file
diff --git a/mountain_car.m b/mountain_car.m
new file mode 100644
index 0000000..c42bac3
--- /dev/null
+++ b/mountain_car.m
@@ -0,0 +1,175 @@
+%% https://rvl.cs.toronto.edu/backwards-reachability/#22-in-two-simple-dimensions
+%% https://github.com/rvl-lab-utoronto/backwards-reachability
+
+%% https://www.cs.ubc.ca/~mitchell/ToolboxLS/
+%% https://github.com/HJReachability/helperOC/blob/master/Intro%20to%20Reachability%20Code.pdf
+
+
+%% https://gieseanw.wordpress.com/2012/10/21/a-comprehensive-step-by-step-tutorial-to-computing-dubins-paths/ 
+
+function mountain_car()
+% 1. Run Backward Reachable Set (BRS) with a goal
+%     uMode = 'min' <-- goal
+%     minWith = 'none' <-- Set (not tube)
+%     compTraj = false <-- no trajectory
+% 2. Run BRS with goal, then optimal trajectory
+%     uMode = 'min' <-- goal
+%     minWith = 'none' <-- Set (not tube)
+%     compTraj = true <-- compute optimal trajectory
+% 3. Run Backward Reachable Tube (BRT) with a goal, then optimal trajectory
+%     uMode = 'min' <-- goal
+%     minWith = 'minVOverTime' <-- Tube (not set)
+%     compTraj = true <-- compute optimal trajectory
+% 5. Change to an avoid BRT rather than a goal BRT
+%     uMode = 'max' <-- avoid
+%     dMode = 'min' <-- opposite of uMode
+%     minWith = 'minVOverTime' <-- Tube (not set)
+%     compTraj = false <-- no trajectory
+% 6. Change to a Forward Reachable Tube (FRT)
+%     add schemeData.tMode = 'forward'
+%     note: now having uMode = 'max' essentially says "see how far I can
+%     reach"
+% 7. Add obstacles
+%     add the following code:
+%     obstacles = shapeCylinder(g, 3, [-1.5; 1.5; 0], 0.75);
+%     HJIextraArgs.obstacles = obstacles;
+% 8. Add random disturbance (white noise)
+%     add the following code:
+%     HJIextraArgs.addGaussianNoiseStandardDeviation = [0; 0; 0.5];
+
+
+%% https://en.wikipedia.org/wiki/Mountain_car_problem
+%% https://github.com/openai/gym/blob/master/gym/envs/classic_control/mountain_car.py
+
+min_velocity = -0.07;
+max_velocity = 0.07;
+
+min_position = -1.2;
+max_position = 0.6;
+
+%% Should we compute the trajectory?
+compTraj = true;
+% compTraj = false;
+
+%% Grid
+grid_min = [min_position; min_velocity]; % Lower corner of computation domain
+grid_max = [max_position; max_velocity];    % Upper corner of computation domain
+N = [181; 181];         % Number of grid points per dimension
+grid = createGrid(grid_min, grid_max, N);
+
+% state space dimensions
+
+%% target set
+% R = 1;
+% data0 = shapeCylinder(grid,ignoreDims,center,radius)
+% making the inital shape 
+% data0 = shapeCylinder(grid, 3, [0; 0; 0], R);
+% also try shapeRectangleByCorners, shapeSphere, etc.
+
+% center = [0.55; 0];
+% widths = [0.05; 0.01];
+
+center = [0.5; 0];
+widths = [0.1; 0.02];
+
+data0 = shapeRectangleByCenter(grid, center, widths);
+
+%% time vector
+t0 = 0;
+% changed the time from 2 seconds to 15
+tMax = 80;
+dt = 0.1;
+tau = t0:dt:tMax;
+
+%% problem parameters
+gravity = 0.0025;
+force = 0.001;
+
+% control trying to min or max value function?
+uMode = 'min';
+
+
+%% Pack problem parameters
+
+% Define dynamic system
+mCar = MountainCarV0([0, 0], gravity, force);
+
+% Put grid and dynamic systems into schemeData
+schemeData.grid = grid;
+schemeData.dynSys = mCar;
+% schemeData.accuracy = 'high'; 
+schemeData.accuracy = 'veryHigh'; 
+schemeData.uMode = uMode;
+
+%% Compute value function
+
+HJIextraArgs.visualize.valueSet = 1;
+HJIextraArgs.visualize.initialValueSet = 1;
+HJIextraArgs.visualize.figNum = 1; %set figure number
+HJIextraArgs.visualize.deleteLastPlot = true; %delete previous plot as you update
+% HJIextraArgs.visualize.valueSetHeatMap = true;
+
+HJIextraArgs.makeVideo = true; % generate video of output
+% You can further customize the video file in the following ways 
+% HJIextraArgs.videoFilename:       (string) filename of video
+% HJIextraArgs.frameRate:           (int) framerate of video
+HJIextraArgs.visualize.xTitle = 'x axis';
+HJIextraArgs.visualize.yTitle = 'y axis';
+
+% uncomment if you want to see a 2D slice
+% HJIextraArgs.visualize.plotData.plotDims = [1 1 0]; %plot x, y
+% HJIextraArgs.visualize.plotData.projpt = [0]; %project at theta = 0
+% HJIextraArgs.visualize.viewAngle = [0,90]; % view 2D
+
+%[data, tau, extraOuts] = ...
+% HJIPDE_solve(data0, tau, schemeData, minWith, extraArgs)
+
+minWith = 'none';
+% minWith = 'maxVOverTime';
+
+[data, tau2, ~] = ...
+  HJIPDE_solve(data0, tau, schemeData, minWith, HJIextraArgs);
+
+%% Compute optimal trajectory from some initial state
+if compTraj
+  pause
+  
+  %set the initial state
+  xinit = [-0.5, 0];
+  
+  figure(6)
+  clf
+  h = visSetIm(grid, data(:,:,:,end));
+%   h.FaceAlpha = .3;
+  hold on
+  s = scatter3(xinit(1), xinit(2), xinit(3));
+  s.SizeData = 70;
+  
+  %check if this initial state is in the BRS/BRT
+  %value = eval_u(g, data, x)
+  value = eval_u(grid,data(:,:,:,end),xinit);
+  
+  if value <= 0 %if initial state is in BRS/BRT
+    % find optimal trajectory
+    
+    mCar.x = xinit; %set initial state of the dubins car
+
+    TrajextraArgs.uMode = uMode; %set if control wants to min or max
+    TrajextraArgs.visualize = true; %show plot
+    TrajextraArgs.fig_num = 2; %figure number
+    
+    %we want to see the first two dimensions (x and y)
+    TrajextraArgs.projDim = [1 1 0]; 
+    
+    %flip data time points so we start from the beginning of time
+    dataTraj = flip(data,4);
+    
+    % [traj, traj_tau] = ...
+    % computeOptTraj(g, data, tau, dynSys, extraArgs)
+    [traj, traj_tau] = ...
+      computeOptTraj(grid, dataTraj, tau2, mCar, TrajextraArgs);
+  else
+    error(['Initial state is not in the BRS/BRT! It have a value of ' num2str(value,2)])
+  end
+end
+end
\ No newline at end of file