Since Q is symmetric you can always find a similarity transformation which will make your Q diagonal. So after the transformation z=T*x the quadratic term x'*Q*x=z'*(T'*Q*T)*z will just be a sum of the squared elements of z with positive weights.

For example if you mainly want to minimize the difference between two state variables, such as 10*(x1-x2)^2+x1^2, then you will get the following matrix Q=[11 -10;-10 10]. This also has negative off-diagonal elements, but are related to the square of the difference between two state variables.

Your example does not have a very nice decomposition, but it still follows the same idea. Namely x'*Q*x will be the sum of the squares of different linear combinations of the full state weighted by some positive number.

Dear Haitham,

“Q” is a weighting matrix that represents the distance (to be minimized across the criterion) between the states (taken separately) that will contribute to the formation of the optimal trajectory. For your case these are distances between pairs (position ', position) and pairs (Velocity', Velocity).

For more details please see attached publications in topics.

Best regards

Article Autonomous Flight of Unmanned Aerial Vehicle (UAV) by using ...

Conference Paper Applicable Values of Turbine Speed Governor Regulation by Li...

Conference Paper DC MOTOR SPEED CONTROL USING DYNAMIC LINEAR QUADRATIC REGULATOR (LQR)

Article Turbofan Engine Modelling and Control Design using Linear Qu...

Dear Haitham,

 As already mentioned  other commentators “Q” is  a symmetric matrix, it is also positive definite, so after appropriate coordinate transform will become a diagonal matrix  representing the weighted sum of squares. In these  coordinates there is no off-diagonal terms and the physical meaning   of each element is   the variance  of appropriate coordinate; the criterion serves to optimize the (weighted) sum of variances. In  the original coordinate the  off diagonal terms  describe the  mutual covariance between coordinates ( in your case between x-position  and y-position/variance, etc...).  There is no surprise that some of them are negative ( as soon as Q remains positive definite).

Beast regards

If you consider that you are using quadratic (matrix) functions, you could give them an energy-based interpretation. Negative values are terms used to substract those corresponding terms. Remember the way you define energy (potential or kinetic energy) with scalar or matrix terms.