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Having a solver for equality constrained QP only will not help you in solving the general inequality constrained problem (unless you develop a QP solver, which can have an equality-constrained QP solver as a required sub-routine). In other words, your problem cannot be reformulated to a problem with only equalities.

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For the Hamiltonian approach you get $$I(x,u,\lambda)=\int_0^1(x^2+xu+u^2+λ(2x+u-\dot x))dt=\int_0^1 (H(x,u,λ)-λ\dot x)\,dt$$ The optimal control then minimizes $H(x,u,λ)$ for fixed $u$ and $λ$. As this is a quadratic convex function, the minimum is easy to compute as $x+2u+λ=0\implies u=-\frac{x+λ}2$. The saddle point can be computed by setting the ...

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Constant Hamiltonian in Optimal Control Theory are related to the Beltrami Identity appearing in Calculus of Variations. In Calculus of Variations, if the Lagrangian $\mathcal{L}$ don't explicetly depend on time such that $J = \int \mathcal{L}(x,\dot{x}) dt$ then the Beltrami Identity $\mathcal{L} - \frac{\partial \mathcal{L}}{\partial \dot{x}_{\alpha}}... 1 I suppose that you mean$y_k=\sin(Cx_k)$, where$Cx$is a scalar. Otherwise, either your$x$is a scalar (and$C=1$), or you should specify the$\sin(x)$function. Ok, so let us define the estimation error signal as$\tilde{x}:=\hat{x}-x$. Suppose that$|\tilde{x}|$is small. Then the following approximation holds$$\sin(Cx)\approx \sin(C\hat{x}) - \cos(C\... 1 As an operations researcher who uses DP, I will say that for the most part, OR treats DP as a decision-making tool, i.e., a method for mathematical optimization, as you say. There are cases in which we try to discern special structure within the solution to the DP. For example, in inventory theory, we use the structure of the DP model to prove that the ... 1 This is quite a subtle matter to my understanding. Also you might specify your question a bit better to get more solid answers. But this vague question allows for creative answers so here you go: (Most of my background are in applied math so I will speak from my perspective.) If you are in the realm of Differential Equations, the rules are many and although ... 1 Your second approach is essentially deadbeat control (driving the output of the system in the least amount of time to the reference), for which it is normal that the generated control inputs are very large and aggressive. Deadbeat control is often very sensitive to how accurate the model has been identified, since it essentially cancels both its pole and ... 2 The choice$Q = C^T C\$ basically puts cost on the system output. Since often, the output of the system is what we want to control, this is a simple choice that can be used as a first try. In practise, however, the states (respectively the outputs) can be on very different scales - therefore, it might not make much sense to weight the outputs like that. ...

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