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First of all, hello and thank you for your time.


Context

I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\limits_{\Omega\subset\mathbb{R}^n}f(\mathbf{x},\mathbf{y}, \mathbf{\nabla y})d\mathbf{x}$$ $$\frac{\partial y_i}{\partial t}=k\Delta\frac{\delta F}{\delta y_i}$$ Where $\Delta$ is the Laplacian and $$\frac{\delta F}{\delta y_i}=\sum\limits_k\frac{\partial}{\partial x_k}\frac{\partial f}{\partial (\partial_ky_i)}-\frac{\partial f}{\partial y_i}$$ is the variational derivative ($\partial _k$ is the partial derivative with respect to $x_k$). Also: $\mathbf{x}\in\mathbb{R}^n, \mathbf{y}\in\mathbb{R}^m$

The system should satisfy the global conservation constraints: $$J_i[\mathbf{y}]=\int\limits_{\Omega\subset\mathbb{R}^n}y_id\mathbf{x}=k_i$$ Where the $k_i$ are constants.

I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $J_i$ increased with time, which was obviously not intended.


Question

I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[\mathbf{y}]=F[\mathbf{y}]-\sum\limits_i\lambda_i J_i[\mathbf{y}]$$

But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):

Suppose that $F$ has an extremum at $y\in C^2[x_0, x_1]$ subject to the boundary conditions [...]. Then there exist two numbers $\lambda_0, \lambda_1$ not both zero such that $$\frac{\delta K}{\delta y}=0$$ Where $K=\lambda_0 F-\lambda_1 J$

As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $\mathbf{y}$ gets to the steady state (i.e. it approaches it asymptotically).

Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $F$ not being stationary?

I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.

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