AI: select immediate acceleration / rotation to reach the end point

Question

I am working on a game where, with every update of the game cycle, AI is executed. During this update, I have a chance to turn an object controlled by AI and / or speed it up in the direction it is. I want it to reach its final location (within a reasonable range) and in this place have a certain speed and direction (again, this does not need to be exact) That is, given the current:

• P ₀ (x, y) = Current Position Vector
• V ₀ (x, y) = Vector of the current speed (units / second)
• θ ₀ = Current direction (radians)
• τ _max = Maximum rotation speed (radian / second)
• α _max = Maximum acceleration (units / second ^ 2)
• | V | _max = Absolute maximum speed (units / second)
• P _f (x, y) = target position vector
• V _f (x, y) = target velocity vector (units / second)
• θ _f = Target rotation (radian)

Choose Immediate:

• τ = Rotational speed within [-τ _max , τ _max ]
• α = Acceleration scalar within [0, α _max ] (should be accelerated in the direction in which it is currently)

Thus, they are minimized:

• t=
• | P _t -P _f |=
• | V _t -V _f |=
• | θ _t -θ _f |= ( (-π, π))

. 1000 , , , , ( - ):

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α τ Δt (Δt → 0 ) / , ( , , , ):

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_{(EDIT: θ = θ 0 + τΔt)}

, α τ (, , > 100 )? , , , - :

• Δt, Δts (.. )
• 5 (α, τ) = {0, α _max} x {-τ _max, 0, τ _max} ( 6 5 , , , )
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, (, , ) , P _f, V _f θ _f (.. / ), .

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Answer 1

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thats

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Answer 2

http://en.wikipedia.org/wiki/Bang%E2%80%93bang_control ( Bang-bang), PID PD. , , , , , , , , , . , - .